Recent zbMATH articles in MSC 37https://zbmath.org/atom/cc/372023-09-22T14:21:46.120933ZWerkzeugOn Polish groups admitting non-essentially countable actionshttps://zbmath.org/1517.030382023-09-22T14:21:46.120933Z"Kechris, Alexander S."https://zbmath.org/authors/?q=ai:kechris.alexander-s"Malicki, Maciej"https://zbmath.org/authors/?q=ai:malicki.maciej"Panagiotopoulos, Aristotelis"https://zbmath.org/authors/?q=ai:panagiotopoulos.aristotelis"Zielinski, Joseph"https://zbmath.org/authors/?q=ai:zielinski.josephSummary: It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.Classification of maps on a finite set under permutationhttps://zbmath.org/1517.050092023-09-22T14:21:46.120933Z"García Zapata, Juan Luis"https://zbmath.org/authors/?q=ai:garcia-zapata.juan-luis"Rico-Gallego, Juan-Antonio"https://zbmath.org/authors/?q=ai:rico-gallego.juan-antonioSummary: We characterize when two maps \(f,g: X \rightarrow X\) on a finite set are conjugated, that is, when there is a permutation \(\sigma: X \rightarrow X\) such that \(f= \sigma^{-1} g \sigma\). We build a signature \(\operatorname{sgn}(f)\) for every map \(f\) such that \(f\) and \(g\) are conjugated if and only if \(\operatorname{sgn}(f)= \operatorname{sgn}(g)\). This signature is an array that includes information about the cycles of the map and the noncyclic elements, called transients, besides data about the insertion of the transients in the cycle. The transient elements form several tree shape graphs. Our characterization is a generalization of the known fact that two permutations are conjugated if and only if they have the same cycle structure. We use elementary facts about finite dynamical systems and about the canonical labeling of unordered trees. The signature can be built by an algorithm of complexity \(O(n)\), being \(n\) the cardinality of \(X\).Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groupshttps://zbmath.org/1517.110972023-09-22T14:21:46.120933Z"Pollicott, M."https://zbmath.org/authors/?q=ai:pollicott.mark"Vytnova, P."https://zbmath.org/authors/?q=ai:vytnova.polinaThis paper is devoted to the exact value of the Hausdorff dimension and to limits sets for some dimensional Markov iterated function schemes. Also, the present research deals with the following topics: Diophantine approximations, the difference between the Markov and Lagrange spectra, and denominators of finite continued fractions and the Zaremba conjecture, as well as the spectrum of the Laplacian on certain Riemann surfaces.
Special attention is given to auxiliary notions, explanations, and examples. Such notions as the Hausdorff dimension, continued fractions, and the Markov spectrum, as well as the Markov iterated function schemes and the transfer operator, etc., are recalled. In addition, the pressure function is considered.
Approaches for estimating the Hausdorff dimension of dynamically defined sets given by Markov iterated function schemes, are discussed. The authors present the other approach which ``is based on combining elements of the methods of \textit{K. I. Babenko} and \textit{S. P. Yur'ev} [Sov. Math., Dokl. 19, 731--735 (1978; Zbl 0416.10040); translation from Dokl. Akad. Nauk SSSR 240, 1273--1276 (1978)] and \textit{E. Wirsing} [Acta Arith. 24, 507--528 (1974; Zbl 0283.10032)] originally developed for the Gauss map''.
Ingredients for effective estimates of the Hausdorff dimension are described. The used techniques and related peculiarities and known results are explained.
Finally, one can note the following text from authors' abstract:
``\dots we are particularly concerned with the role of the value of the Hausdorff dimension in solving conjectures and problems in other areas of mathematics. As our first application we confirm, and often strengthen, conjectures on the difference of the Lagrange and Markov spectra in Diophantine analysis, which appear in the work of \textit{C. Matheus} and \textit{C. G. Moreira} [Comment. Math. Helv. 95, No. 3, 593--633 (2020; Zbl 1465.11165)]. As a second application we (re-)validate and improve estimates connected with the Zaremba conjecture in number theory, used in the work of \textit{J. Bourgain} and \textit{A. Kontorovich} [Ann. Math. (2) 180, No. 1, 137--196 (2014; Zbl 1370.11083)], \textit{S. Huang} [Geom. Funct. Anal. 25, No. 3, 860--914 (2015; Zbl 1333.11078)] and \textit{I. D. Kan} [Sb. Math. 210, No. 3, 364--416 (2019; Zbl 1437.11010); translation from Mat. Sb. 210, No. 3, 75--130 (2019)]. As a third more geometric application, we rigorously bound the bottom of the spectrum of the Laplacian for infinite area surfaces, as illustrated by an example studied by \textit{C. T. McMullen} [Am. J. Math. 120, No. 4, 691--721 (1998; Zbl 0953.30026)]\dots''
Reviewer: Symon Serbenyuk (Kyjiw)Cluster algebras and higher order generalizations of the \(q\)-Painleve equations of type \(A_7^{(1)}\) and \(A_6^{(1)}\)https://zbmath.org/1517.130192023-09-22T14:21:46.120933Z"Masuda, Tetsu"https://zbmath.org/authors/?q=ai:masuda.tetsu"Okubo, Naoto"https://zbmath.org/authors/?q=ai:okubo.naoto"Tsuda, Teruhisa"https://zbmath.org/authors/?q=ai:tsuda.teruhisaSummary: We construct higher order generalizations of the \(q\)-Painlevé equations of surface type \(A_7^{(1)}\) and \(A_6^{(1)}\) based on the cluster algebras corresponding to certain quivers. These equations possess the affine Weyl group symmetries of type \(A_1^{(1)}\) and \((A_1+A'_1)^{(1)}\), respectively. We show that these equations and symmetries can be realized as birational canonical transformations. A relationship between the quivers and the discrete KdV equation is also discussed.Discrete parametric surfaceshttps://zbmath.org/1517.140242023-09-22T14:21:46.120933Z"Wallner, Johannes"https://zbmath.org/authors/?q=ai:wallner.johannes|wallner.johannes-peterSummary: Discrete parametric surfaces are discrete analogues of smooth parametric surfaces. They are, however, not simply discrete approximations of their smooth counterparts, but are the subject of a separate discrete theory. As it turns out, a systematic theory of parametric surfaces can be based on integrable systems, and the discrete case can be interpreted as a ``master'' case which contains smooth surfaces as a limit.
This section on discrete parametric surfaces is organized as follows: We first introduce notation. Two particular kinds of discrete surfaces are discussed next: circular nets in \S2, and K-nets in \S3 are examples of a 3-system and a 2-system, respectively. In the case of K-nets, we also discuss the relation to the sine-Gordon equation. We then show applications within mathematics in \S4, cf. [\textit{A. I. Bobenko} et al., Ann. Math. (2) 164, No. 1, 231--264 (2006; Zbl 1122.53003)], and the connection with freeform architecture in \S5, cf. [\textit{H. Pottmann} and \textit{J. Wallner}, in: Mathematics and society. Zürich: European Mathematical Society (EMS). 131--151 (2016; Zbl 1354.00063)]. The main source for this chapter is the monograph [\textit{A. I. Bobenko} and \textit{Y. B. Suris}, Discrete differential geometry. Integrable structure. Providence, RI: American Mathematical Society (AMS) (2008; Zbl 1158.53001)].
For the entire collection see [Zbl 1455.53045].Locally random groupshttps://zbmath.org/1517.220032023-09-22T14:21:46.120933Z"Mallahi-Karai, Keivan"https://zbmath.org/authors/?q=ai:karai.keivan-mallahi"Mohammadi, Amir"https://zbmath.org/authors/?q=ai:mohammadi.amir"Salehi Golsefidy, Alireza"https://zbmath.org/authors/?q=ai:salehi-golsefidy.alirezaSummary: In this work, we introduce and study the notion of \textit{local randomness} for compact metric groups. We prove a mixing inequality as well as a product result for locally random groups under an additional \textit{dimension condition} on the volume of small balls, and provide several examples of such groups. In particular, this leads to new examples of groups satisfying such a mixing inequality. In the same context, we develop a Littlewood-Paley decomposition and explore its connection to the existence of a spectral gap for random walks. Moreover, under the dimension condition alone, we prove a multi-scale entropy gain result à la Bourgain-Gamburd and Tao.Fundamental groups, 3-braids, and effective estimates of invariantshttps://zbmath.org/1517.300032023-09-22T14:21:46.120933Z"Jöricke, Burglind"https://zbmath.org/authors/?q=ai:joricke.burglindThe paper is devoted to invariants of braids and their effective bounds. For a subset \(A\) of \(\mathbb C\), consider the configuration space \(C_n(A)=\{(z_1,\dots,z_n)\in A^n:z_i\neq z_j\;\text{for}\;i\neq j\}\) of \(n\) particles moving along \(A\) without collision. For a symmetric group \(S_n\), the quotient \(C_n(A)/S_n\) is the symmetrized configuration space related to \(A\). Choose a base point \(E_n\in C_n(\mathbb C)/S_n\) and regard \(n\)-braids as homotopy classes of loops in the symmetrized configuration space, equivalently, as elements of the fundamental group \(\pi_1(C_n(\mathbb C)/S_n,E_n)\) of the symmetrized configuration space with base point \(E_n\).
The author defines the extremal lengths for a braid \(b\) in the group \(\mathcal B_3\) of 3-braids and the conformal module of \(b\) with totally real horizontal boundary values. For a 3-braid, there is a definition of the extremal length of \(b\) with perpendicular bisector boundary values. In the paper, upper and lower bounds are given for the versions of the extremal length of any braid. Also, the estimates give bounds for the entropy of pure 3-braids.
Reviewer: Dmitri V. Prokhorov (Saratov)Counting arcs on hyperbolic surfaceshttps://zbmath.org/1517.320302023-09-22T14:21:46.120933Z"Bell, Nick"https://zbmath.org/authors/?q=ai:bell.nickSummary: We give the asymptotic growth of the number of arcs of bounded length between boundary components on hyperbolic surfaces with boundary. Specifically, if \(S\) has genus \(g, n\) boundary components and \(p\) punctures, then the number of orthogeodesic arcs in each pure mapping class group orbit of length at most \(L\) is asymptotic to \(L^{6g - 6+2 (n+p)}\) times a constant. We prove an analogous result for arcs between cusps, where we define the length of such an arc to be the length of the sub-arc obtained by removing certain cuspidal regions from the surface.Lacunary series, resonances, and automorphisms of \({\mathbb{C}^2}\) with a round Siegel domainhttps://zbmath.org/1517.320362023-09-22T14:21:46.120933Z"Berteloot, François"https://zbmath.org/authors/?q=ai:berteloot.francois"Cheraghi, Davoud"https://zbmath.org/authors/?q=ai:cheraghi.davoudSummary: We construct transcendental automorphims of \({\mathbb{C}^2}\) that have an unbounded and regular Siegel domain.A lower bound for the Hausdorff dimension of the Green currenthttps://zbmath.org/1517.320372023-09-22T14:21:46.120933Z"de Thélin, Henry"https://zbmath.org/authors/?q=ai:de-thelin.henryLet \(f\) be a birational self-map of a compact Kähler surface \(X\), and \(T^\pm\) be the Green currents of \(f\) and \(f^{-1}\). If \(f\) satisfies the so-called Bedford-Diller condition, then the measure \(\mu:=T^+\wedge T^-\) is well-defined and is hyperbolic with one positive and one negative Lyapunov exponent. The main result of this paper (Theorem 1) shows that if the first dynamical degree of \(f\) is greater than one then the Hausdorff dimension of the support of the Green current \(T^+\) is greater than two.
Theorem 1 follows from a more general result (Theorem 2) which is true in any dimension, while Theorem 2 is a consequence of Theorem 3 which is proven using Pesin's theory and Oseledets' theorem.
Reviewer: Feng Rong (Shanghai)Periodic points of weakly post-critically finite all the way down mapshttps://zbmath.org/1517.320382023-09-22T14:21:46.120933Z"Van Tu Le"https://zbmath.org/authors/?q=ai:van-tu-le.1Summary: We study eigenvalues along periodic cycles of post-critically finite endomorphisms of \(\mathbb{CP}^n\) in higher dimension. It is a classical result when \(n = 1\) that those values are either 0 or of modulus strictly bigger than 1. It has been conjectured in [the author, Ergodic Theory Dyn. Syst. 42, No. 7, 2382--2414 (2022; Zbl 1495.32052)] that the same result holds for every \(n \ge 2\). In this article, we verify the conjecture for the class of weakly post-critically finite all the way down maps which was introduced in [\textit{M. Astorg}, Ergodic Theory Dyn. Syst. 40, No. 2, 289--308 (2020; Zbl 1437.37059)]. This class contains a well-known class of post-critically finite maps constructed in [\textit{S. Koch}, Adv. Math. 248, 573--617 (2013; Zbl 1310.32016)]. As a consequence, we verify the conjecture for Koch maps.Regularity of the equilibrium measure for meromorphic correspondenceshttps://zbmath.org/1517.320412023-09-22T14:21:46.120933Z"Dinh, Tien-Cuong"https://zbmath.org/authors/?q=ai:tien-cuong-dinh."Wu, Hao"https://zbmath.org/authors/?q=ai:wu.hao.8Summary: Let \(f\) be a meromorphic correspondence on a compact Kähler manifold \(X\) of dimension \(k\). Assume that its topological degree is larger than the dynamical degree of order \(k-1\). We obtain a quantitative regularity of the equilibrium measure of \(f\) in terms of its super-potentials.Nonexistence of invariant manifolds in fractional-order dynamical systemshttps://zbmath.org/1517.340032023-09-22T14:21:46.120933Z"Bhalekar, Sachin"https://zbmath.org/authors/?q=ai:bhalekar.sachin"Patil, Madhuri"https://zbmath.org/authors/?q=ai:patil.madhuri(no abstract)Phase portraits of planar piecewise linear refracted systems: node-saddle casehttps://zbmath.org/1517.340412023-09-22T14:21:46.120933Z"Wang, Yidan"https://zbmath.org/authors/?q=ai:wang.yidan"Wei, Zhouchao"https://zbmath.org/authors/?q=ai:wei.zhouchao"Liu, Haozhe"https://zbmath.org/authors/?q=ai:liu.haozhe"Zhang, Wei"https://zbmath.org/authors/?q=ai:zhang.wei.7Summary: Global phase portraits can reveal the long-term dynamical behavior and the presence of special dynamical phenomena. For the piecewise linear (PWL) refracted systems, the previous researches include node-node case, saddle-saddle case, focus-focus case, etc. This paper investigates the global dynamics of node-saddle PWL refracted systems with a linear separation line. The focus is on the dynamical behavior of homogeneous orbits, limit cycles, etc., arising from non-smooth properties. The main results include not only the existence and stability of limit cycles and homoclinic orbits in bounded areas but also the qualitative proposition of equilibria at infinity by using Liénard canonical form. Finally, this paper gives a schematic representation of the possibilities of all topologically inequivalent global phase portraits of these refracted systems using Poincaré disc.Homoclinic bifurcation of limit cycles in near-Hamiltonian systems on the cylinderhttps://zbmath.org/1517.340522023-09-22T14:21:46.120933Z"Shi, Yixia"https://zbmath.org/authors/?q=ai:shi.yixia"Han, Maoan"https://zbmath.org/authors/?q=ai:han.maoan"Zhang, Lijun"https://zbmath.org/authors/?q=ai:zhang.lijun.2Summary: In this paper, we study a near-Hamiltonian system on the cylinder. First, we establish some general methods on the existence of limit cycles bifurcating from closed orbits of type II by the Melnikov function method, then we derive the expansions of the first order Melnikov function and consider the bifurcation problem of limit cycles near a double homoclinic loop. As an application, we discuss the number of limit cycles of a class of cylinder pendulum-like systems.Measuring the criticality of a Hopf bifurcationhttps://zbmath.org/1517.340562023-09-22T14:21:46.120933Z"Uteshev, Alexei"https://zbmath.org/authors/?q=ai:uteshev.alexei-yu"Kalmár-Nagy, Tamás"https://zbmath.org/authors/?q=ai:kalmar-nagy.tamas(no abstract)Characterization of the Kukles polynomial differential systems having an invariant algebraic curvehttps://zbmath.org/1517.340622023-09-22T14:21:46.120933Z"Llibre, Jaume"https://zbmath.org/authors/?q=ai:llibre.jaume"Valls, Claudia"https://zbmath.org/authors/?q=ai:valls.claudiaThis paper characterizes all Kukles polynomial differential systems of the form
\[
\dot x= y, \dot y= -y^2-f(x)y- g(x)
\]
having an invariant algebraic curve. The authors prove that expanding this invariant algebraic curve as a polynomial in the variable \(y\), the first four higher coefficients of the polynomial defining the invariant algebraic curve determine completely these Kukles systems.
Reviewer: Man Jia (Fuzhou)Transmission dynamics and high infectiousness of coronavirus disease 2019https://zbmath.org/1517.340672023-09-22T14:21:46.120933Z"Huang, Shunxiang"https://zbmath.org/authors/?q=ai:huang.shunxiang"Wu, Lin"https://zbmath.org/authors/?q=ai:wu.lin"Li, Jing"https://zbmath.org/authors/?q=ai:li.jing.38|li.jing.6|li.jing.5|li.jing.4|li.jing.2|li.jing.10|li.jing.1|li.jing.3|li.jing.7|li.jing.19|li.jing.17|li.jing.11|li.jing.15|li.jing|li.jing.67|li.jing.12|li.jing.32|li.jing.33|li.jing.13|li.jing.16"Xin, Ming-Zhen"https://zbmath.org/authors/?q=ai:xin.mingzhen"Wang, Yingying"https://zbmath.org/authors/?q=ai:wang.yingying|wang.yingying.1"Hao, Xingjie"https://zbmath.org/authors/?q=ai:hao.xingjie"Wang, Zhongyi"https://zbmath.org/authors/?q=ai:wang.zhongyi"Deng, Qihong"https://zbmath.org/authors/?q=ai:deng.qihong"Wang, Bin-Guo"https://zbmath.org/authors/?q=ai:wang.binguoSummary: Coronavirus disease 2019 (COVID-19) has rapidly spread around the world since the early 2020. Recently, a second wave of COVID-19 has resurged in many countries. The transmission dynamics and infectiousness of the COVID-19 pandemic remain unclear, and developing strategies to mitigate the severity of the pandemic is a top priority for global public health. According to the infection mechanism of COVID-19, a novel susceptible-asymptomatic-symptomatic-recovered (SASR) model with control variables in a patchy environment was proposed not only to consider the key characteristics of asymptomatic infection and the effects of seasonal variation but also to incorporate different control measures for multiple transmission routes. The basic reproduction number \(R_0\) was established to describe the spreading behavior in the natural state over a long time horizon, and the natural reproduction number \(R_n\), which describes the development trend of the disease during a short time in the future, was defined according to the actual propagation characteristics. In addition, the effective reproduction number \(R_e\) considering the control strategies was proposed to evaluate the impact of non-pharmaceutical interventions. The results of numerical simulations for COVID-19 cases in Wuhan, China, based on the SASR model indicate that \(R_0\) was 3.58, \(R_n\) ranged from 2.37 to 4.91, and \(R_e\) decreased gradually from 4.83 on December 8, 2019 to 0.31 on March 8, 2020, reaching 1.40 on January 23, 2020, when the lockdown was lifted in Wuhan. We further concluded that the total number of infections, including asymptomatic infections, was approximately 301, 804 as of March 8, 2020, in Wuhan, China. In particular, this article proposes a dynamic method to distinguish the impact of natural factors and human interventions on the development of the pandemic, and provides a theoretical basis for fighting the global COVID-19 pandemic.Existence and stability of periodic solutions for a mosquito suppression model with incomplete cytoplasmic incompatibilityhttps://zbmath.org/1517.340762023-09-22T14:21:46.120933Z"Yan, Rong"https://zbmath.org/authors/?q=ai:yan.rong"Zheng, Bo"https://zbmath.org/authors/?q=ai:zheng.bo.1"Yu, Jianshe"https://zbmath.org/authors/?q=ai:yu.jian-sheSummary: Cytoplasmic incompatibility (CI) caused by endosymbiotic bacteria \textit{Wolbachia} has been an effective way to suppress and even eradicate wild vector mosquitoes. By assuming that CI is driven by impulsive and periodic releases of \textit{Wolbachia}-infected males, we develop a time-switching ordinary differential equation model to characterize the impact of the release amount, the release period and the incomplete CI on the suppression efficiency. Sufficient conditions to guarantee the existence and uniqueness, or the existence of exactly two periodic solutions are obtained, together with their stability analyses.Formal stability, stability for most initial conditions and diffusion in analytic systems of differential equationshttps://zbmath.org/1517.340792023-09-22T14:21:46.120933Z"Kozlov, Valery V."https://zbmath.org/authors/?q=ai:kozlov.valerii-vasilievichSummary: An example of an analytic system of differential equations in \(\mathbb{R}^6\) with an equilibrium formally stable and stable for most initial conditions is presented. By means of a divergent formal transformation this system is reduced to a Hamiltonian system with three degrees of freedom. Almost all its phase space is foliated by three-dimensional invariant tori carrying quasi-periodic trajectories. These tori do not fill all phase space. Though the ``gap'' between these tori has zero measure, this set is everywhere dense in \(\mathbb{R}^6\) and unbounded phase trajectories are dense in this gap. In particular, the formally stable equilibrium is Lyapunov unstable. This behavior of phase trajectories is quite consistent with the diffusion in nearly integrable systems. The proofs are based on the Poincaré - Dulac theorem, the theory of almost periodic functions, and on some facts from the theory of inhomogeneous Diophantine approximations. Some open problems related to the example are presented.On the stability of linear time-varying differential equationshttps://zbmath.org/1517.340802023-09-22T14:21:46.120933Z"Zaitsev, V. A."https://zbmath.org/authors/?q=ai:zaitsev.vasili-alexandrovich"Kim, I. G."https://zbmath.org/authors/?q=ai:kim.inna-geraldovna|kim.inna-geraldovaSummary: The article discusses the stability of linear differential equations with time-varying coefficients. It is shown that, in contrast to equations with time-invariant coefficients, the condition for the characteristic polynomial to be Hurwitz for a linear differential equation with time-varying coefficients is neither necessary nor sufficient for the asymptotic stability of the differential equation. It is proved that the analog of Kharitonov's theorem on robust stability does not hold if the coefficients of the differential equation are time-varying.The Lorenz system: hidden boundary of practical stability and the Lyapunov dimensionhttps://zbmath.org/1517.340812023-09-22T14:21:46.120933Z"Kuznetsov, N. V."https://zbmath.org/authors/?q=ai:kuznetsov.nikolay-v"Mokaev, T. N."https://zbmath.org/authors/?q=ai:mokaev.timur-nazirovich"Kuznetsova, O. A."https://zbmath.org/authors/?q=ai:kuznetsova.olga-aleksandrovna"Kudryashova, E. V."https://zbmath.org/authors/?q=ai:kudryashova.elena-vladimirovna(no abstract)Chaos transition of the generalized fractional Duffing oscillator with a generalized time delayed position feedbackhttps://zbmath.org/1517.340872023-09-22T14:21:46.120933Z"El-Borhamy, Mohamed"https://zbmath.org/authors/?q=ai:el-borhamy.mohamed(no abstract)Homoclinic solutions for \(n\)-dimensional prescribed mean curvature \(p\)-Laplacian equationshttps://zbmath.org/1517.340932023-09-22T14:21:46.120933Z"Lu, Shiping"https://zbmath.org/authors/?q=ai:lu.shiping"Kong, Fanchao"https://zbmath.org/authors/?q=ai:kong.fanchaoSummary: In this paper, a \(n\)-dimensional prescribed mean curvature Rayleigh \(p\)-Laplacian equation with a deviating argument, \((\varphi_{p}(\frac{u'(t)}{\sqrt{1+| u'(t)|^{2}}}))'+F(t,u'(t))+G(t,u(t-\tau(t)))=e(t)\), is studied. By means of Mawhin's continuation theorem and some analysis methods, a~new result on the existence of homoclinic solutions for the equation is obtained. Our research enriches the contents of prescribed mean curvature equations.The direct method of Lyapunov for nonlinear dynamical systems with fractional dampinghttps://zbmath.org/1517.340962023-09-22T14:21:46.120933Z"Hinze, Matthias"https://zbmath.org/authors/?q=ai:hinze.matthias"Schmidt, André"https://zbmath.org/authors/?q=ai:schmidt.andre"Leine, Remco I."https://zbmath.org/authors/?q=ai:leine.remco-i(no abstract)A novel chaotic map constructed by geometric operations and its applicationhttps://zbmath.org/1517.340992023-09-22T14:21:46.120933Z"Zhang, Zhiqiang"https://zbmath.org/authors/?q=ai:zhang.zhiqiang"Wang, Yong"https://zbmath.org/authors/?q=ai:wang.yong.10"Zhang, Leo Yu"https://zbmath.org/authors/?q=ai:zhang.leo-yu"Zhu, Hong"https://zbmath.org/authors/?q=ai:zhu.hong.1(no abstract)The safety problem for nonlinear systems with delay in terms of barrier functionshttps://zbmath.org/1517.341032023-09-22T14:21:46.120933Z"Sedova, N. O."https://zbmath.org/authors/?q=ai:sedova.n-oSummary: We study the safety problem for nonlinear nonautonomous systems described by the Cauchy problem for differential equation with delay with ordinary derivative. To analyze safety, we introduce the Razumikhin barrier function that guarantees the invariance of some safety set. Unlike traditional barrier functions used in optimization problems, our construction is not unbounded on the boundary of an admissible domain, but only changes the sign. Owing to this fact, we use the barrier function to construct a control providing the safety of the system. The results are illustrated by examples.On the delayed vector-bias malaria model in an almost periodic environmenthttps://zbmath.org/1517.341102023-09-22T14:21:46.120933Z"He, Bing"https://zbmath.org/authors/?q=ai:he.bing.1|he.bing|he.bing.4|he.bing.2|he.bing.3"Qiang, Lizhong"https://zbmath.org/authors/?q=ai:qiang.lizhongSummary: The global dynamics of a vector-bias malaria model with extrinsic incubation period in an almost periodic environment are considered. The basic reproduction ratio \(R_0\) is first introduced, and then it is proved that \(R_0\) is a threshold parameter that determines the global dynamics of the model, that is, the disease-free almost periodic solution is globally attractive when \(R_0 <1\), the system allows a unique positive almost periodic solution which has global attractivity when \(R_0 >1\). Numerical simulations show that extending the incubation period is advantageous for disease control, and the basic reproduction ratio may be overestimated or underestimated if vector bias is ignored.Bi-space global attractors for a class of second-order evolution equations with dispersive and dissipative terms in locally uniform spaceshttps://zbmath.org/1517.350592023-09-22T14:21:46.120933Z"Zhang, Fang-hong"https://zbmath.org/authors/?q=ai:zhang.fanghongSummary: This paper deals with the asymptotic behavior of a class of second-order evolution equations with dispersive and dissipative terms' critical nonlinearity in locally uniform spaces. First of all, we prove the global well-posedness of solutions to the evolution equations in the locally uniform spaces \(H^1_{\mathrm{lu}}(\mathbb{R}^N)\times H^1_{\mathrm{lu}}(\mathbb{R}^N)\) and define a strong continuous analytic semigroup. Secondly, the existence of the \((H^1_{\mathrm{lu}}(\mathbb{R}^N)\times H^1_{\mathrm{lu}}(\mathbb{R}^N), H^1_\rho(\mathbb{R}^N)\times H^1_\rho(\mathbb{R}^N))\)-global attractor is established. Finally, we obtain the asymptotic regularity of solutions which appear to be optimal and the existence of a bounded subset(in \(H^2_{\mathrm{lu}}(\mathbb{R}^N)\times H^2_{\mathrm{lu}}(\mathbb{R}^N)\)), which attracts exponentially every initial \(H^1_{\mathrm{lu}}(\mathbb{R}^N)\times H^1_{\mathrm{lu}}(\mathbb{R}^N)\)-bounded set with respect to the \(H^1_{\mathrm{lu}}(\mathbb{R}^N)\times H^1_{\mathrm{lu}}(\mathbb{R}^N)\)-norm.Multi-mode solitons in a long-short range traffic lattice model with time delayhttps://zbmath.org/1517.350892023-09-22T14:21:46.120933Z"Ren, Xiufang"https://zbmath.org/authors/?q=ai:ren.xiufang"Zhao, Shiji"https://zbmath.org/authors/?q=ai:zhao.shiji(no abstract)Antidark solitons and soliton molecules in a \((3 + 1)\)-dimensional nonlinear evolution equationhttps://zbmath.org/1517.350912023-09-22T14:21:46.120933Z"Wang, Xin"https://zbmath.org/authors/?q=ai:wang.xin.15"Wei, Jiao"https://zbmath.org/authors/?q=ai:wei.jiao(no abstract)Random attractors for non-autonomous stochastic Navier-Stokes-Voigt equations in some unbounded domainshttps://zbmath.org/1517.351612023-09-22T14:21:46.120933Z"Wang, Shu"https://zbmath.org/authors/?q=ai:wang.shu"Si, Mengmeng"https://zbmath.org/authors/?q=ai:si.mengmeng"Yang, Rong"https://zbmath.org/authors/?q=ai:yang.rongSummary: This paper is concerned with the asymptotic behavior for the three dimensional non-autonomous stochastic Navier-Stokes-Voigt equations on unbounded domains. A continuous non-autonomous random dynamical system for the equations is firstly established. We then obtain pullback asymptotic compactness of solutions and prove that the existence of tempered random attractors for the random dynamical system generated by the equations. Furthermore, we obtain that the tempered random attractors are periodic when the deterministic non-autonomous external term is periodic in time.Entropy estimates for uniform attractors of 2D Navier-Stokes equations with weakly normal measureshttps://zbmath.org/1517.351622023-09-22T14:21:46.120933Z"Xiong, Yangmin"https://zbmath.org/authors/?q=ai:xiong.yangmin"Song, Xiaoya"https://zbmath.org/authors/?q=ai:song.xiaoya"Sun, Chunyou"https://zbmath.org/authors/?q=ai:sun.chunyouSummary: This paper aims at the long-time behavior of non-autonomous 2D Navier-Stokes equations with a class of external forces which are \(H\)-valued measures in time. We first establish the well-posedness of solutions as well as the existence of a strong uniform attractor, and then pay the main attention on the estimation of \(\varepsilon\)-entropy for such uniform attractor in the standard energy phase space.Parametric analysis of dust ion acoustic waves in superthermal plasmas through non-autonomous KdV frameworkhttps://zbmath.org/1517.351932023-09-22T14:21:46.120933Z"Chadha, Naresh M."https://zbmath.org/authors/?q=ai:chadha.naresh-m"Tomar, Shruti"https://zbmath.org/authors/?q=ai:tomar.shruti"Raut, Santanu"https://zbmath.org/authors/?q=ai:raut.santanuSummary: In the presence of superthermal plasma, propagation of non-linear dust ion acoustic waves has been studied in the framework of the Damped Forced Korteweg-de Vries (DFKdV) Equation. A feasible range is obtained for the existence of the solitary wave solutions in terms of the spectral index, and the unaffected dust-to-ion density ratio (denoted by \(\kappa\) and \(\mu\), respectively). It is observed that the transition of solitary wave structures from compressive to rarefactive can be completely determined by mainly these two parameters. The effects of all other parameters involved in the model, namely, the strength \((f_0)\) and frequency \((\omega)\) of the external periodic force, and damping coefficient \((\nu_{\mathrm{id}0})\) are studied using the bifurcation analysis; \(f_0\) and \(\omega\) play a crucial role in the periodic and chaotic motions in the system. We also obtained certain critical values of the key parameters involved in the model for which the model exhibits periodic, quasi-periodic and chaotic behaviour.A focusing-defocusing intermediate nonlinear Schrödinger systemhttps://zbmath.org/1517.352002023-09-22T14:21:46.120933Z"Berntson, Bjorn K."https://zbmath.org/authors/?q=ai:berntson.bjorn-k"Fagerlund, Alexander"https://zbmath.org/authors/?q=ai:fagerlund.alexanderSummary: We introduce and study a system of coupled nonlocal nonlinear Schrödinger equations that interpolates between the mixed, focusing-defocusing Manakov system on one hand and a limiting case of the intermediate nonlinear Schrödinger equation on the other. We show that this new system, which we call the intermediate mixed Manakov (IMM) system, admits multi-soliton solutions governed by a complexification of the hyperbolic Calogero-Moser (CM) system. Furthermore, we introduce a spatially periodic version of the IMM system, for which our main result is a class of exact solutions governed by a complexified elliptic CM system.Multi-soliton solutions for a higher-order coupled nonlinear Schrödinger system in an optical fiber via Riemann-Hilbert approachhttps://zbmath.org/1517.352042023-09-22T14:21:46.120933Z"Guo, Han-Dong"https://zbmath.org/authors/?q=ai:guo.handong"Xia, Tie-Cheng"https://zbmath.org/authors/?q=ai:xia.tie-cheng(no abstract)Rogue wave and multi-pole solutions for the focusing Kundu-Eckhaus equation with nonzero background via Riemann-Hilbert problem methodhttps://zbmath.org/1517.352052023-09-22T14:21:46.120933Z"Guo, Ning"https://zbmath.org/authors/?q=ai:guo.ning|guo.ning.1"Xu, Jian"https://zbmath.org/authors/?q=ai:xu.jian.1"Wen, Lili"https://zbmath.org/authors/?q=ai:wen.lili"Fan, Engui"https://zbmath.org/authors/?q=ai:fan.engui(no abstract)Continuum limit for the Ablowitz-Ladik systemhttps://zbmath.org/1517.352072023-09-22T14:21:46.120933Z"Killip, Rowan"https://zbmath.org/authors/?q=ai:killip.rowan"Ouyang, Zhimeng"https://zbmath.org/authors/?q=ai:ouyang.zhimeng"Visan, Monica"https://zbmath.org/authors/?q=ai:visan.monica"Wu, Lei"https://zbmath.org/authors/?q=ai:wu.lei.1|wu.lei.4|wu.lei.2|wu.lei.3|wu.leiSummary: We show that solutions to the Ablowitz-Ladik system converge to solutions of the cubic nonlinear Schrödinger equation for merely \(L^2\) initial data. Furthermore, we consider initial data for this lattice model that excites Fourier modes near both critical points of the discrete dispersion relation and demonstrate convergence to a \textit{decoupled} system of nonlinear Schrödinger equations.Stable and oscillating solitons of \(\mathcal{PT}\)-symmetric couplers with gain and loss in fractional dimensionhttps://zbmath.org/1517.352162023-09-22T14:21:46.120933Z"Zeng, Liangwei"https://zbmath.org/authors/?q=ai:zeng.liangwei"Shi, Jincheng"https://zbmath.org/authors/?q=ai:shi.jincheng"Lu, Xiaowei"https://zbmath.org/authors/?q=ai:lu.xiaowei"Cai, Yi"https://zbmath.org/authors/?q=ai:cai.yi"Zhu, Qifan"https://zbmath.org/authors/?q=ai:zhu.qifan"Chen, Hongyi"https://zbmath.org/authors/?q=ai:chen.hongyi"Long, Hu"https://zbmath.org/authors/?q=ai:long.hu"Li, Jingzhen"https://zbmath.org/authors/?q=ai:li.jingzhen(no abstract)Increasing stability in the inverse scattering problem for a nonlinear Schrödinger equation with multiple frequencieshttps://zbmath.org/1517.352172023-09-22T14:21:46.120933Z"Zhao, Yue"https://zbmath.org/authors/?q=ai:zhao.yueSummary: This paper is concerned with the inverse scattering problem of determining the unknown coefficients for a nonlinear two-dimensional Schrödinger equation. We establish for the first time the increasing stability of the inverse scattering problem from the multi-frequency far-field pattern for nonlinear equations. To achieve this goal, we prove the existence of a holomorphic region and an upper bound for the solution with respect to the complex wavenumber, which also leads to the well-posedness of the direct scattering problem. The stability estimate consists of the Lipschitz type data discrepancy and the high frequency tail of the unknown coefficients, where the latter decreases as the upper bound of the frequency increases.Averaging principle for stochastic complex Ginzburg-Landau equationshttps://zbmath.org/1517.352182023-09-22T14:21:46.120933Z"Cheng, Mengyu"https://zbmath.org/authors/?q=ai:cheng.mengyu"Liu, Zhenxin"https://zbmath.org/authors/?q=ai:liu.zhenxin"Röckner, Michael"https://zbmath.org/authors/?q=ai:rockner.michaelSummary: Averaging principle is an effective method for investigating dynamical systems with highly oscillating components. In this paper, we study three types of averaging principle for stochastic complex Ginzburg-Landau equations. Firstly, we prove that the solution of the original equation converges to that of the averaged equation on finite intervals as the time scale \(\varepsilon\) goes to zero when the initial data are the same. Secondly, we show that there exists a unique recurrent solution (in particular, periodic, almost periodic, almost automorphic, etc.) to the original equation in a neighborhood of the stationary solution of the averaged equation when the time scale is small. Finally, we establish the global averaging principle in weak sense, i.e. we show that the attractor of original system tends to that of the averaged equation in probability measure space as \(\varepsilon\) goes to zero.Resonant double Hopf bifurcation in a diffusive Ginzburg-Landau model with delayed feedbackhttps://zbmath.org/1517.352202023-09-22T14:21:46.120933Z"Huang, Yuxuan"https://zbmath.org/authors/?q=ai:huang.yuxuan"Zhang, Hua"https://zbmath.org/authors/?q=ai:zhang.hua.5"Niu, Ben"https://zbmath.org/authors/?q=ai:niu.ben(no abstract)On Lax equations of the two-component BKP hierarchyhttps://zbmath.org/1517.352742023-09-22T14:21:46.120933Z"Geng, Lumin"https://zbmath.org/authors/?q=ai:geng.lumin"Hu, Jianxun"https://zbmath.org/authors/?q=ai:hu.jianxun"Wu, Chao-Zhong"https://zbmath.org/authors/?q=ai:wu.chaozhongSummary: We propose a Lax representation of the two-component BKP hierarchy via pseudo-differential operators containing two derivations, and show that it is equivalent to the Lax representation via two types of pseudo-differential operators involving only one derivation given by \textit{S.-Q. Liu} et al. [Int. Math. Res. Not. 2011, No. 8, 1952--1996 (2011; Zbl 1221.35458)].Notes on Hamiltonian dynamical systemshttps://zbmath.org/1517.370012023-09-22T14:21:46.120933Z"Giorgilli, Antonio"https://zbmath.org/authors/?q=ai:giorgilli.antonioPublisher's description: Starting with the basics of Hamiltonian dynamics and canonical transformations, this text follows the historical development of the theory culminating in recent results: the Kolmogorov-Arnold-Moser theorem, Nekhoroshev's theorem and superexponential stability. Its analytic approach allows students to learn about perturbation methods leading to advanced results. Key topics covered include Liouville's theorem, the proof of Poincaré's non-integrability theorem and the nonlinear dynamics in the neighbourhood of equilibria. The theorem of Kolmogorov on persistence of invariant tori and the theory of exponential stability of Nekhoroshev are proved via constructive algorithms based on the Lie series method. A final chapter is devoted to the discovery of chaos by Poincaré and its relations with integrability, also including recent results on superexponential stability. Written in an accessible, self-contained way with few prerequisites, this book can serve as an introductory text for senior undergraduate and graduate students.Introduction to Arnold's proof of the Kolmogorov-Arnold-Moser theoremhttps://zbmath.org/1517.370022023-09-22T14:21:46.120933Z"Feldmeier, Achim"https://zbmath.org/authors/?q=ai:feldmeier.achimThe purpose of this book is to present a detailed and step-by-step proof of the classic KAM theorem (named after Kolmogorov, Arnold and Moser) that fills in many of the details that are sketched in Arnold's originally published proof. The theorem asserts the existence of invariant Lagrangian tori for nearly integrable systems. The tori are quasi-periodic with Diophantine frequency vectors. Their union is a nowhere dense set of positive measure in the phase space. The KAM theorem shows that periodic, deterministic motion (as for example in celestial mechanics) is stable under sufficiently small perturbations for all times.
The KAM theorem was first stated by Kolmogorov in 1954, proved in a somewhat different form by Moser in 1962, and then by Arnold in 1963. Arnold's original published proof had barely 15 pages. While it uses essentially only standard methods of real and complex analysis and the individual steps of the proof are elementary, it applies the methods in intricate ways. Since Arnold's original proof can be essentially impenetrable even to scientists with a solid mathematical background, the author of this book (a physicist) makes the effort to fill in the details carefully. He considers the KAM theorem as one of the classics of the 20th century mathematics, and believes that Arnold's proof offers one of the best contributions to KAM theory. As a physicist he also prefers Arnold's proof as most relevant to describing a real Hamiltonian mechanical system.
After background and preliminaries, the author presents an outline of Arnold's proof. He describes, in general terms, the main ideas and techniques. These include more traditional things like perturbation theory, some ideas relatively new in Arnold's time (such as super-convergence, sets without interior, and Diophantine conditions to deal with small divisor problems) as well as new methods like the cutoff of Fourier series. The main proof follows; it consists of a fundamental theorem, an inductive theorem, and the KAM theorem itself.
Provided with the basic proof are four additional chapters that contain needed auxiliary lemmas: these are roughly characterized as analytic, geometric, convergence and arithmetic. Although the proof truly uses only standard techniques of analysis (with a little bit of degree theory), putting it all together is an intricate process. The author has done an excellent job in assembling and explaining how all the pieces fit together.
An excellent bibliography is offered to the reader as well as a a collection of references to previous reviews of the KAM theorem and its proofs.
Reviewer: William J. Satzer Jr. (St. Paul)Groups, languages and dendric shiftshttps://zbmath.org/1517.370032023-09-22T14:21:46.120933Z"Perrin, Dominique"https://zbmath.org/authors/?q=ai:perrin.dominiqueSummary: We present a survey of results obtained on symbolic dynamical systems called dendric shifts. We state and sketch the proofs (sometimes new ones) of the main results obtained on these shifts. This includes the return theorem and the finite index basis theorem which both put in evidence the central role played by free groups in these systems. We also present a series of applications of these results, including some on profinite semigroups and some on dimension groups.
For the entire collection see [Zbl 1398.68030].On the pullback relation on curves induced by a Thurston maphttps://zbmath.org/1517.370042023-09-22T14:21:46.120933Z"Pilgrim, Kevin M."https://zbmath.org/authors/?q=ai:pilgrim.kevin-mThis is a survey on the dynamics of Thurston maps via their action on curves on the sphere minus the postcritical set.
A Thurston map is an orientation-preserving branched covering of the sphere \(f: S^2 \to S^2\) of degree at least two with the property that the forward orbit of its critical set \(P_f:= \bigcup_{n > 0} f^n(C_f)\) is finite. Here, \(C_f\) is the finite set of points where \(f\) is not locally injective. These maps have been intensively studied and include the class of postcritically finite rational maps of the Riemann sphere.
Given a Thurston map \(f\) with postcritical set \(P_f\), we can pullback a simple closed curve \(\gamma\) in \(S^2 \setminus P_f\) by \(f\) and take one of its components \(\widetilde \gamma\). The main object of study of this survey is the dynamical behavior of iterated pre-images of simple closed curves (up to isotopy) under a given Thurston map.
The author presents many examples and known results together with open questions and conjectures about the pullback relation. For example, it is conjectured that if \(f\) is a postcritically finite rational map that is not a flexible Lattès example, then the pullback relation on curves has a finite global attractor. A list of partial answers to this conjecture is given. The author also discusses some analogies with the theory of mapping class groups.
For the entire collection see [Zbl 1495.57001].
Reviewer: Lucas Kaufmann (Orléans)Directional Kronecker algebra for \(\mathbb{Z}^q\)-actionshttps://zbmath.org/1517.370052023-09-22T14:21:46.120933Z"Liu, Chunlin"https://zbmath.org/authors/?q=ai:liu.chunlin"Xu, Leiye"https://zbmath.org/authors/?q=ai:xu.leiyeIn this paper, the directional sequence entropy and the directional Kronecker algebra for \(\mathbb{Z}^q\)-systems are introduced. The relation between sequence entropy and directional sequence entropy is established. Meanwhile, directional discrete spectrum systems and directional null systems are defined. It is shown that a \(\mathbb{Z}^q\)-system has directional discrete spectrum if and only if it is directional null. Moreover, it turns out that a \(\mathbb{Z}^q\)-system has directional discrete spectrum along \(q\) linearly independent directions if and only if it has discrete spectrum.
Reviewer: Daniele Puglisi (Catania)An uncountable ergodic Roth theorem and applicationshttps://zbmath.org/1517.370062023-09-22T14:21:46.120933Z"Durcik, Polona"https://zbmath.org/authors/?q=ai:durcik.polona"Greenfeld, Rachel"https://zbmath.org/authors/?q=ai:greenfeld.rachel"Iseli, Annina"https://zbmath.org/authors/?q=ai:iseli.annina"Jamneshan, Asgar"https://zbmath.org/authors/?q=ai:jamneshan.asgar"Madrid, José"https://zbmath.org/authors/?q=ai:madrid.jose-a-jimenezAmenable group actions, Følner sequences, and recurrence theorems are fundamental concepts in the study of group actions and dynamics. An amenable group is a group that possess certain properties, such as the existence of a left-invariant mean on its left regular representation. Følner sequences, on the other hand, are sequences of finite subsets of a group that exhibit a balanced growth property. These sequences play a crucial role in establishing key results in the theory of amenable group actions, particularly in proving various forms of recurrence theorems.
The paper studies extensions, uniformity and combinatorial implications related to Furstenberg's double recurrence theorem. This theorem, which is closely connected to Roth's theorem, plays a central role throughout the paper.
The authors prove an uncountable version of the ergodic Roth theorem of \textit{V. Bergelson} et al. [Am. J. Math. 119, No. 6, 1173--1211 (1997; Zbl 0886.43002)]
for discrete amenable groups. More specifically, they show that if \(\Gamma\) is an arbitrary amenable discrete group then for every Følner net a certain ergodic average in an abstract Roth \(\Gamma\)-dynamical system converges, and, this situation is independent of the choice of the Følner net. The group is not assumed to be countable, and the space is not necessarily separable. Then the authors use this theorem to obtain a syndetic subset of \(\Gamma.\) Recall that a subset of a group or semigroup is said to be syndetic if it has bounded gaps between its elements. Syndetic sets are closely related to multiple recurrence and multiple ergodic averages. Their presence or absence has important implications for the behavior of ergodic averages and the convergence of related theorems. The syndetic subset obtained by the authors are used for the syndeticity of multiple return times for an ultralimit system. By using an uncountable version of the Furstenberg correspondence principle they prove a combinatorial theorem about syndetic sets and triangular patterns in certain subsets of the Cartesian square of arbitrary amenable groups.
One of the main results of the paper is the syndeticity of double return times for arbitrary amenable groups by extending the double recurrence theorem of \textit{V. Bergelson} et al. [loc. cit.] to uncountable amenable groups acting on arbitrary not necessarily separable spaces. Further they obtain a new uniformity aspect for the set of double return times in the amenable Roth theorem. They generalize this to Roth-type \(\Gamma\)-measure-preserving dynamical systems where \(\Gamma\) is uniformly amenable. They show the existence of a lower bound on the degree of syndeticity uniformly over a class of Roth-type measure-preserving dynamical systems for a uniformly amenable set of groups. To do this they use lower Banach densities defined on subsets of the discrete group \(\Gamma\) and they consider ultraproducts of measure-preserving dynamical systems.
Reviewer: Nazife Erkursun Ozcan (Ankara)Entropy, products, and bounded orbit equivalencehttps://zbmath.org/1517.370072023-09-22T14:21:46.120933Z"Kerr, David"https://zbmath.org/authors/?q=ai:kerr.david"Li, Hanfeng"https://zbmath.org/authors/?q=ai:li.hanfengSummary: We prove that if two topologically free and entropy regular actions of countable sofic groups on compact metrizable spaces are continuously orbit equivalent, and each group either (i) contains a w-normal amenable subgroup which is neither locally finite nor virtually cyclic, or (ii) is a non-locally-finite product of two infinite groups, then the actions have the same sofic topological entropy. This fact is then used to show that if two free uniquely ergodic and entropy regular probability-measure-preserving actions of such groups are boundedly orbit equivalent then the actions have the same sofic measure entropy. Our arguments are based on a relativization of property SC to sofic approximations and yield more general entropy inequalities.Subgroups of continuous full groups and relative continuous orbit equivalences of one-sided topological Markov shiftshttps://zbmath.org/1517.370082023-09-22T14:21:46.120933Z"Matsumoto, Kengo"https://zbmath.org/authors/?q=ai:matsumoto.kengoSummary: Let \(A\) be an irreducible non-permutation matrix with entries in \(\{0, 1\}\). We study a family \(\Gamma_{A,f}\) of subgroups of the continuous full group \(\Gamma_A\) of the one-sided topological Markov shift \((X_A, \sigma_A)\). The subgroup \(\Gamma_{A,f}\) is indexed by an integer valued continuous function \(f\) on \(X_A\) such that \(\Gamma_{A, 0} = \Gamma_A\) the non-amenable continuous full group and \(\Gamma_{A,1} = \Gamma_A^{\mathrm{AF}}\) the amenable AF full group. We will first prove that a spatial realization theorem for the subgroups \(\Gamma_{A,f}\) under certain conditions on the functions \(f\). We will second introduce relative versions of continuous orbit equivalence to study classification of the subgroups \(\Gamma_{A,f}\) related to their associated groupoids and \(C^{\ast}\)-algebras.Ergodic theorem in CAT(0) spaces in terms of inductive meanshttps://zbmath.org/1517.370092023-09-22T14:21:46.120933Z"Antezana, Jorge"https://zbmath.org/authors/?q=ai:antezana.jorge"Ghiglioni, Eduardo"https://zbmath.org/authors/?q=ai:ghiglioni.eduardo-m"Stojanoff, Demetrio"https://zbmath.org/authors/?q=ai:stojanoff.demetrioSummary: Let \((G,+)\) be a compact, abelian, and metrizable topological group. In this group we take \(g\in G\) such that the corresponding automorphism \(\tau_g\) is ergodic. The main result of this paper is a new ergodic theorem for functions in \(L^1(G,M)\), where \(M\) is a Hadamard space. The novelty of our result is that we use inductive means to average the elements of the orbit \(\{\tau_g^n(h)\}_{n\in\mathbb{N}}\). The advantage of inductive means is that they can be explicitly computed in many important examples. The proof of the ergodic theorem is done firstly for continuous functions, and then it is extended to \(L^1\) functions. The extension is based on a new construction of mollifiers in Hadamard spaces. This construction has the advantage that it only uses the metric structure and the existence of barycenters, and does not require the existence of an underlying vector space. For this reason, it can be used in any Hadamard space, in contrast to those results that need to use the tangent space or some chart to define the mollifier.A view on multiple recurrencehttps://zbmath.org/1517.370102023-09-22T14:21:46.120933Z"Eisner, Tanja"https://zbmath.org/authors/?q=ai:eisner.tanjaSummary: In this note we present a proof of multiple recurrence for ergodic systems (and thereby of Szemerédi's theorem) being a mixture of three known proofs. It is based on a conditional version of the Jacobs-de Leeuw-Glicksberg decomposition and properties of the Gowers-Host-Kra uniformity seminorms.Ergodicity of Iwasawa continued fractions via markable hyperbolic geodesicshttps://zbmath.org/1517.370112023-09-22T14:21:46.120933Z"Lukyanenko, Anton"https://zbmath.org/authors/?q=ai:lukyanenko.anton"Vandehey, Joseph"https://zbmath.org/authors/?q=ai:vandehey.josephSummary: We prove the convergence and ergodicity of a wide class of real and higher-dimensional continued fraction algorithms, including folded and \(\alpha\)-type variants of complex, quaternionic, octonionic, and Heisenberg continued fractions, which we combine under the framework of Iwasawa continued fractions. The proof is based on the interplay of continued fractions and hyperbolic geometry, the ergodicity of geodesic flow in associated modular manifolds, and a variation on the notion of geodesic coding that we refer to as geodesic marking. As a corollary of our study of markable geodesics, we obtain a generalization of Serret's tail-equivalence theorem for almost all points. The results are new even in the case of some real and complex continued fractions.Some counterexamples to the central limit theorem for random rotationshttps://zbmath.org/1517.370122023-09-22T14:21:46.120933Z"Czudek, Klaudiusz"https://zbmath.org/authors/?q=ai:czudek.klaudiuszWhen we talk about Central Limit Theorems (CLTs) in probability theory, we, often quite understandably, focus on the success stories and provide sufficient conditions that ensure a Gaussian limit. The present paper by is a refreshing take on the subject as it provides a number of counterexamples to celebrated CLTs.
The primary object of study is a discrete-time Random Walk (RW) \((Y_n^\alpha)_{n\ge 1}\) on the circle \(\mathbb{S}^{1}\). At each time step, the random walk moves to either \(x+\alpha\) or \(x-\alpha\) with equal probability, where \(\alpha \in \mathbb{R}\). One should think of \(\alpha\) as the angle of rotation. The author is able to connect properties of the angle with the validity of the CLT for additive functionals of the random walk, i.e., if \(n^{-1/2}(\psi(Y_1^\alpha) + \psi(Y_2^\alpha) + \ldots + \psi(Y_n^\alpha))\) converges to a Gaussian distribution. In particular, the angle \(\alpha\) is said to be Diophantine of type \((c, \gamma) \in (0, \infty)\times[2, \infty)\) if \(|\alpha - p/q | \ge c q^{-\gamma}\) for all integers \(p, q \ne 0\). It is called Liouville if it is not Diophantine.
The authors prove the following:
(1) There exists a Liouville angle and an analytic functional for which the CLT does not hold;
(2) If the angle \(\alpha\) is irrational and satisfies \(|\alpha - p/q| \le cq^{-\gamma}\) for infinitely many integers \(p, q\ne 0\) and some \(c>0, \gamma >5\), then there exists a continuous function with first \(r\) derivatives, where \(r\) is the largest positive integer less than \((\gamma - 3)/2\);
(3) For a given Liouville angle, there exist smooth functions \(\psi\) such that the CLT does not hold.
The results obtained in this paper are reminiscent of conjugacy results for circle diffeomorphisms. The proof goes by solving the Poisson equation, which is a functional analytic approach, but has also become standard in probability theory.
The results presented in this paper are refreshing and leave you with uncanny satisfaction!
Reviewer: Wasiur Rahman Khuda Bukhsh (Nottingham)On certain inverse semigroups associated with one-sided topological Markov shiftshttps://zbmath.org/1517.370132023-09-22T14:21:46.120933Z"Matsumoto, Kengo"https://zbmath.org/authors/?q=ai:matsumoto.kengoSummary: We will introduce an inverse semigroup written \({\mathcal{S}}_A\) associated with a one-sided topological Markov shift \((X_A,\sigma_A)\) for an irreducible matrix \(A\) with entries in \(\{0,1\}\). We will show that two inverse semigroups \({\mathcal{S}}_A\) and \({\mathcal{S}}_B\) are isomorphic if and only if the one-sided topological Markov shifts \((X_A,\sigma_A)\) and \((X_B, \sigma_B)\) are continuous orbit equivalent. As a result, the isomorphism class of the inverse semigroup \({\mathcal{S}}_A\) is described in terms of an étale groupoid \(G_A\), Cuntz-Krieger algebra \({\mathcal{O}}_A\), and so on.On \(C^0\)-genericity of distributional chaoshttps://zbmath.org/1517.370142023-09-22T14:21:46.120933Z"Kawaguchi, Noriaki"https://zbmath.org/authors/?q=ai:kawaguchi.noriakiSummary: Let \(M\) be a compact smooth manifold without boundary. Based on results by \textit{C. Good} and \textit{J. Meddaugh} [Invent. Math. 220, No. 3, 715--736 (2020; Zbl 1445.37015)], we prove that a strong distributional chaos is \(C^0\)-generic in the space of continuous self-maps (respectively, homeomorphisms) of \(M\). The results contain answers to questions by \textit{J. Li} et al. [Chaos 26, No. 9, 093103, 6 p. (2016; Zbl 1382.37034)] and \textit{T. K. S. Moothathu} [Topology Appl. 158, No. 16, 2232--2239 (2011; Zbl 1235.54018)] in the zero-dimensional case. A related counter-example on the chain components under shadowing is also given.Generic properties of homeomorphisms preserving a given dynamical simplexhttps://zbmath.org/1517.370152023-09-22T14:21:46.120933Z"Melleray, Julien"https://zbmath.org/authors/?q=ai:melleray.julienSummary: Given a dynamical simplex \(K\) on a Cantor space \(X\), we consider the set \(G_K^*\) of all homeomorphisms of \(X\) which preserve all elements of \(K\) and have no non-trivial clopen invariant subset. Generalizing a theorem of \textit{A. Q. Yingst} [Ergodic Theory Dyn. Syst. 41, No. 10, 3178--3200 (2021; Zbl 1494.37003)], we prove that for a generic element \(g\) of \(G_K^*\) the set of invariant measures of \(g\) is equal to \(K\). We also investigate when there exists a generic conjugacy class in \(G_K^*\) and prove that this happens exactly when \(K\) has only one element, which is the unique invariant measure associated to some odometer; and that in that case the conjugacy class of this odometer is generic in \(G_K^*\).Independence and almost automorphy of higher orderhttps://zbmath.org/1517.370162023-09-22T14:21:46.120933Z"Qiu, Jiahao"https://zbmath.org/authors/?q=ai:qiu.jiahaoSummary: In this paper it is shown that for a minimal system \((X,T)\) and \(d,k\in\mathbb{N}\), if \((x,x_i)\) is regionally proximal of order \(d\) for \(1\leq i\leq k\), then \((x,x_1,\dots,x_k)\) is \((k+1)\)-regionally proximal of order \(d\). Meanwhile, we introduce the notion of \(\mathrm{IN}^{[d]}\)-pair: for a dynamical system \((X,T)\) and \(d\in\mathbb{N}\), a pair \((x_0,x_1)\in X\times X\) is called an \(\mathrm{IN}^{[d]}\)-pair if for any \(k\in\mathbb{N}\) and any neighborhoods \(U_0,U_1\) of \(x_0\) and \(x_1\) respectively, there exist different \((p_1^{(i)},\dots,p_d^{(i)})\in \mathbb{N}^d\), \(1\leq i\leq k\), such that
\[
\bigcup\limits_{i=1}^k\{p_1^{(i)}\epsilon(1)+\cdots+p_d^{(i)} \epsilon(d):\epsilon(j)\in\{0,1\},1\leq j\leq d\}\backslash\{0\}\in\mathrm{Ind}(U_0,U_1),
\]
where \(\mathrm{Ind}(U_0,U_1)\) denotes the collection of all independence sets for \((U_0,U_1)\). It turns out that for a minimal system, if it does not contain any non-trivial \(\mathrm{IN}^{[d]}\)-pair, then it is an almost one-to-one extension of its maximal factor of order \(d\).The nonexistence of expansive polycyclic group actions on the circle \(\mathbb{S}^1\)https://zbmath.org/1517.370172023-09-22T14:21:46.120933Z"Shi, Enhui"https://zbmath.org/authors/?q=ai:shi.enhui"Wang, Suhua"https://zbmath.org/authors/?q=ai:wang.suhua"Xie, Zhiwen"https://zbmath.org/authors/?q=ai:xie.zhiwen"Xu, Hui"https://zbmath.org/authors/?q=ai:xu.huiSummary: We show that the circle \(\mathbb{S}^1\) admits no expansive polycyclic group actions.Upper capacity entropy and packing entropy of saturated sets for amenable group actionshttps://zbmath.org/1517.370182023-09-22T14:21:46.120933Z"Zhang, Wenda"https://zbmath.org/authors/?q=ai:zhang.wenda"Ren, Xiankun"https://zbmath.org/authors/?q=ai:ren.xiankun"Zhang, Yiwei"https://zbmath.org/authors/?q=ai:zhang.yiweiSummary: Let \((X,G)\) be a \(G\)-action topological system, where \(G\) is a countable infinite discrete amenable group and \(X\) a compact metric space. In this paper we study the upper capacity entropy and packing entropy for systems with weaker version of specification. We prove that the upper capacity always carries full entropy while there is a variational principle for packing entropy of saturated sets.On periodic decompositions and nonexpansive lineshttps://zbmath.org/1517.370192023-09-22T14:21:46.120933Z"Colle, Cleber Fernando"https://zbmath.org/authors/?q=ai:colle.cleber-fernandoThe author provides some new results based on a conjecture by \textit{M. Szabados} in his Ph.D. thesis [An algebraic approach to Nivat's conjecture. University of Turku (PhD Thesis) (2018)]. In his Ph.D. thesis, Michal Szabados demonstrated how to decompose every low pattern complexity configuration into a finite sum of periodic configurations. Let \(\mathcal{A}\) be an alphabet with at least two elements and \(\mathcal{A}^{\mathbb{Z}^{2}}\) be the space of all configurations of the form \(\eta=(\eta_\textbf{g})_{\textbf{g}\in \mathbb{Z}^{2}}\), where \(\eta_\textbf{g}\in \mathcal{A}\) for all \(\textbf{g}\in \mathbb{Z}^{2}\). Given \(n\in \mathbb{N}\), let us set \([[n]] = \{0, 1, 2, \cdots , n-1\}\). For the configuration \(\eta\in \mathcal{A}^{\mathbb{Z}^{2}}\), let NEL(\(\eta\)) denote the set formed by the nonexpansive lines on \(\overline{\mathrm{Orb}(\eta)}\). The author proves that if NEL\((\eta) =\emptyset\) then \(\eta\) is fully periodic.
The author also proves the following result:
Theorem. Let \(\eta\in \mathcal{A}^{\mathbb{Z}^{2}}\), with \(\mathcal{A} \subset\mathbb{Z}_+\), be a configuration and suppose \(\eta= \eta_1+ \cdots+\eta_m\) is a \(\mathbb{Z}\)-minimal periodic decomposition. Then, for some prime number \(p\in\mathbb{N}\), with \(\mathcal{A}\subset[[p]], \overline{\eta}= \overline{\eta}_1+\cdots+\overline{\eta}_m\) is a \(\mathbb{Z}_p\)-minimal periodic decomposition.
With this paper, the author shows that the conjecture of Michal Szabados holds true for any fully periodic configuration having low convex pattern complexity.
Reviewer: Hasan Akin (Şanlıurfa)Majority rule cellular automatahttps://zbmath.org/1517.370202023-09-22T14:21:46.120933Z"Gärtner, Bernd"https://zbmath.org/authors/?q=ai:gartner.bernd"Zehmakan, Ahad N."https://zbmath.org/authors/?q=ai:n-zehmakan.ahadSummary: Consider a graph \(G=(V,E)\) and a random initial coloring where each vertex is black independently with probability \(p_b\), and white with probability \(p_w=1-p_b\). In each step, all vertices change their current color synchronously to the most frequent color in their neighborhood and in case of a tie, a vertex keeps its current color. This model is called the \textit{majority model}. If in case of a tie a vertex always selects black color, it is called the \textit{biased majority model}. We are interested in the behavior of these two processes, especially when the underlying graph is a two-dimensional torus (cellular automaton with (biased) majority rule). In the present paper, as our main result we prove that both majority and biased majority cellular automata exhibit a threshold behavior with two phase transitions. More precisely, we prove for a two-dimensional torus \(T_{n , n} \), there are two threshold values \(0\leq p_1,p_2\leq 1\) such that \(p_b\ll p_1,p_1\ll p_b\ll p_2\), and \(p_2\ll p_b\) result in final complete occupancy by white, stable coexistence of both colors, and final complete occupancy by black, respectively in \(\mathcal{O}(n^2)\) number of steps. (For two functions \(f(n)\) and \(g(n)\), we shortly write \(f(n)\ll g(n)\) instead of \(f(n)\in o(g(n))\).) We finally argue that our proof techniques can be used to prove a similar threshold behavior for a larger class of models.Rapid left expansivity, a commonality between Wolfram's rule 30 and powers of \(p/q\)https://zbmath.org/1517.370212023-09-22T14:21:46.120933Z"Kopra, Johan"https://zbmath.org/authors/?q=ai:kopra.johanSummary: We define the class of rapidly left expansive cellular automata, which contains Wolfram's Rule 30, fractional multiplication automata, and many others. Previous results on aperiodicity of columns in space-time diagrams of certain cellular automata generalize to this new class. We also present conditions that imply periodic behavior in cellular automata and use these to prove new results on rapidly left expansive cellular automata that originate from the theory of distribution modulo 1.Force recurrence of semigroup actionshttps://zbmath.org/1517.370222023-09-22T14:21:46.120933Z"Yan, Kesong"https://zbmath.org/authors/?q=ai:yan.kesong"Zeng, Fanping"https://zbmath.org/authors/?q=ai:zeng.fanping"Tian, Rong"https://zbmath.org/authors/?q=ai:tian.rongSummary: We investigate the sets of countable discrete semigroups that force recurrence, that is, the recurrent properties of a point along a subset of a countable semigroup action. We show that a subset of a monoid forces recurrence (resp., forces minimality) if and only if it contains a broken \textit{IP}-set (resp., broken syndetic set), and forces infinite recurrence implies it is contains a broken infinite \textit{IP}-sets. As an example, we show that every subset with positive upper Banach density of infinite countable amenable groups forces infinite recurrence.Covering action on Conley index theoryhttps://zbmath.org/1517.370232023-09-22T14:21:46.120933Z"Lima, D. V. S."https://zbmath.org/authors/?q=ai:lima.dahisy-valadao-de-souza"R. Da Silveira, M."https://zbmath.org/authors/?q=ai:r-da-silveira.m"Vieira, E. R."https://zbmath.org/authors/?q=ai:vieira.ewerton-rSummary: In this paper we apply Conley index theory in a covering space of an invariant set \(S\), possibly not isolated, in order to describe the dynamics in \(S\). More specifically, we consider the action of the covering translation group in order to define a topological separation of \(S\) which distinguishes the connections between the Morse sets within a Morse decomposition of \(S\). The theory developed herein generalizes the classical connection matrix theory, since one obtains enriched information on the connection maps for non-isolated invariant sets, as well as for isolated invariant sets. Moreover, in the case of an infinite cyclic covering induced by a circle-valued Morse function, one proves that the Novikov differential of \(f\) is a particular case of the \(p\)-connection matrix defined herein.Topological sequence entropy and topological dynamics of tree mapshttps://zbmath.org/1517.370242023-09-22T14:21:46.120933Z"Cánovas, Jose S."https://zbmath.org/authors/?q=ai:canovas.jose-s"Daghar, Aymen"https://zbmath.org/authors/?q=ai:daghar.aymenSummary: We prove that a zero topological entropy continuous tree map always displays zero topological sequence entropy when it is restricted to its non-wandering and chain recurrent sets. In addition, we show that a similar result is not possible when the phase space is a dendrite even when we consider only the restriction on the set of periodic points.Dendrites and measures with discrete spectrumhttps://zbmath.org/1517.370252023-09-22T14:21:46.120933Z"Foryś-Krawiec, Magdalena"https://zbmath.org/authors/?q=ai:forys-krawiec.magdalena"Hantáková, Jana"https://zbmath.org/authors/?q=ai:hantakova.jana"Kupka, Jiří"https://zbmath.org/authors/?q=ai:kupka.jiri"Oprocha, Piotr"https://zbmath.org/authors/?q=ai:oprocha.piotr"Roth, Samuel"https://zbmath.org/authors/?q=ai:roth.samuelSummary: We are interested in dendrites for which all invariant measures of zero-entropy mappings have discrete spectrum, and we prove that this holds when the closure of the endpoint set of the dendrites is countable. This solves an open question which has been around for awhile, and almost completes the characterization of dendrites with this property.Sharing a measure of maximal entropy in polynomial semigroupshttps://zbmath.org/1517.370262023-09-22T14:21:46.120933Z"Pakovich, Fedor"https://zbmath.org/authors/?q=ai:pakovich.fedorSummary: Let \(P_1,P_2,\dots,P_k\) be complex polynomials of degree at least two that are not simultaneously conjugate to monomials or to Chebyshev polynomials, and \(S\) the semigroup under composition generated by \(P_1,P_2,\dots,P_k\). We show that all elements of \(S\) share a measure of maximal entropy if and only if the intersection of principal left ideals \(SP_1\cap SP_2\cap\dots\cap SP_k\) is non-empty.Zero entropy and stable rotation sets for monotone recurrence relationshttps://zbmath.org/1517.370272023-09-22T14:21:46.120933Z"Qin, Wen-Xin"https://zbmath.org/authors/?q=ai:qin.wenxin"Shen, Bai-Nian"https://zbmath.org/authors/?q=ai:shen.bai-nian"Sun, Yi-Lin"https://zbmath.org/authors/?q=ai:sun.yi-lin"Zhou, Tong"https://zbmath.org/authors/?q=ai:zhou.tongSummary: In this paper, we show that each element in the convex hull of the rotation set of a compact invariant chain transitive set is realized by a Birkhoff solution, which is an improvement of the fundamental lemma of \textit{T. Zhou} and \textit{W.-X. Qin} [Math. Z. 297, No. 3--4, 1673--1692 (2021; Zbl 1465.37024)]
in the study of rotation sets for monotone recurrence relations. We then investigate the properties of rotation sets assuming the system has zero topological entropy. The rotation set for a Birkhoff recurrence class is a singleton and the forward and backward rotation numbers are identical for each solution in the same Birkhoff recurrence class. We also show the continuity of rotation numbers on the set of non-wandering points. If the rotation set is upper-stable, then we show that each boundary point is a rational number, and we also obtain a result of bounded deviation.On the induced measure-theoretic entropy for random dynamical systemshttps://zbmath.org/1517.370282023-09-22T14:21:46.120933Z"Yang, Kexiang"https://zbmath.org/authors/?q=ai:yang.kexiang"Chen, Ercai"https://zbmath.org/authors/?q=ai:chen.ercai"Zhou, Xiaoyao"https://zbmath.org/authors/?q=ai:zhou.xiaoyaoThis paper introduces the concepts of induced topological entropy and induced measure-theoretic entropy for random dynamical systems.
The measure-theoretic entropy of a measurable transformation was presented by \textit{A. N. Kolmogorov} [Dokl. Akad. Nauk SSSR 119, 861--864 (1958; Zbl 0083.10602)] and \textit{Ya. G. Sinai} [Dokl. Akad. Nauk SSSR 124, 768--771 (1959; Zbl 0086.10102)]. The idea of entropy was transferred to the realm of topological dynamical systems by \textit{R. L. Adler} et al. [Trans. Am. Math. Soc. 114, 309--319 (1965; Zbl 0127.13102)] and, using a different (but equivalent) definition, by \textit{R. Bowen} [Trans. Am. Math. Soc. 153, 401--414 (1971; Zbl 0212.29201)]. Then \textit{A. Katok} [Publ. Math., Inst. Hautes Étud. Sci. 51, 137--173 (1980; Zbl 0445.58015)]
provided another equivalent definition of measure-theoretic entropy for an ergodic Borel probability measure and consequently defined a topological version of measure-theoretic entropy of a continuous map on a compact metric space.
Random dynamical systems were described by \textit{L. Arnold} [Random dynamical systems. Berlin: Springer (1998; Zbl 0906.34001)] and a research of entropy in random dynamical systems started afterwords. The notions of measure-theoretic and topological entropy of a continuous bundle random dynamical systems (RDS) appeared.
Inspired by the previous research, this paper proposes the definitions of induced topological entropy and induced measure-theoretic entropy of RDS over \((\Omega,\mathcal{F},\mathbf{P},\vartheta)\). ``Induced'' modifications of entropy are related to \(\varphi\in L^1_{\Omega\times X}(\Omega,C(X))\) with \(\inf \varphi>0\).
Basic properties of the induced topological entropy are deduced and on this basis, a Katok entropy formula is established for induced measure-theoretic entropy of RDS, which shows the relation between induced measure-theoretic and measure-theoretic entropy of RDS. The obtained formula extends a related expression for deterministic dynamical systems. For an investigation of measure-theoretic entropy, the Shannon-McMillan-Breiman theorem of RDS is extensively used.
Finally, a measure-theoretic entropy for induced pointwise dimensions of RDS is studied and a formula for lower and upper induced pointwise dimensions of RDS is derived. This generalizes a result by \textit{L. Barreira} and \textit{J. Schmeling} [Isr. J. Math. 116, 29--70 (2000; Zbl 0988.37029)]
to the induced version of entropy.
Reviewer: Pavel Ludvík (Olomouc)Entropies for factor maps of amenable group actionshttps://zbmath.org/1517.370292023-09-22T14:21:46.120933Z"Zhang, Guohua"https://zbmath.org/authors/?q=ai:zhang.guohua.2|zhang.guohua.1"Zhu, Lili"https://zbmath.org/authors/?q=ai:zhu.liliSummary: In this paper we study various entropies for factor maps of amenable group actions. We prove firstly theorem inequalities linking relative topological entropy and conditional topological entropy (for factor maps of amenable group actions) without any additional assumption, which strengthens conditional variational principles [Nonlinearity 34, No. 8, 5163--5185 (2021; Zbl 1475.37024); Theorems 2.12 and 3.9] proved by \textit{Z. Changrong} under additional assumptions. Then along the line of [\textit{M. Misiurewicz}, Stud. Math. 55, 175--200 (1976; Zbl 0355.54035)], we introduce a new invariant called relative topological tail entropy and prove a Ledrappier's type variational principle concerning it (for factor maps of amenable group actions); consequently, any factor map with zero relative topological tail entropy admits invariant measures with maximal relative entropy, which provides a nontrivial sufficient condition for the existence of invariant measures with maximal relative entropy in the setting of factor maps of amenable group actions.Observable Lyapunov irregular sets for planar piecewise expanding mapshttps://zbmath.org/1517.370302023-09-22T14:21:46.120933Z"Nakano, Yushi"https://zbmath.org/authors/?q=ai:nakano.yushi"Soma, Teruhiko"https://zbmath.org/authors/?q=ai:soma.teruhiko"Yamamoto, Kodai"https://zbmath.org/authors/?q=ai:yamamoto.kodaiSummary: For any integer \(r\) with \(1\le r<\infty\), we present a one-parameter family \(F_\sigma(0<\sigma<1)\) of 2-dimensional piecewise \(\mathcal{C}^r\) expanding maps such that each \(F_\sigma\) has an observable (i.e. Lebesgue positive) Lyapunov irregular set. These maps are obtained by modifying the piecewise expanding map given in [\textit{M. Tsujii}, Ergodic Theory Dyn. Syst. 20, No. 6, 1851--1857 (2000; Zbl 0992.37018)]. In strong contrast to it, we also show that any Lyapunov irregular set of any 2-dimensional piecewise real analytic expanding map is not observable. This is based on the spectral analysis of piecewise expanding maps in [\textit{J. Buzzi}, Ergodic Theory Dyn. Syst. 20, No. 3, 697--708 (2000; Zbl 0973.37003)].Measure preserving diffeomorphisms of the torus are unclassifiablehttps://zbmath.org/1517.370312023-09-22T14:21:46.120933Z"Foreman, Matthew"https://zbmath.org/authors/?q=ai:foreman.matthew-d|foreman.matthew-r"Weiss, Benjamin"https://zbmath.org/authors/?q=ai:weiss.benjamin-l|weiss.benjamin.1|weiss.benjaminThe isomorphism problem in ergodic theory was formulated by \textit{J. von Neumann} [Ann. Math. (2) 33, 587--642 (1932; Zbl 0005.12203)]. The problem has been solved for some classes of transformations that have special properties.
Starting in the late 1990s a different type of result began to appear: anti-classification results that demonstrate in a rigorous way that classification is not possible. The first anti-classification result is due to \textit{F. Beleznay} and \textit{M. Foreman} [Ergodic Theory Dyn. Syst. 16, No. 5, 929--962 (1996; Zbl 0869.58032)] who showed that the class of measure distal transformations used in early ergodic theoretic proofs of Szemeredi's theorem is not a Borel set.
\textit{M. Foreman} et al. [Ann. Math. (2) 173, No. 3, 1529--1586 (2011; Zbl 1243.37006)] have shown that determining isomorphism between ergodic transformations is inaccessible to countable methods that use countable amounts of information.
The purpose of the present paper is to show that the variety of ergodic transformations that have smooth models is rich enough so that the abstract isomorphism relation, when restricted to these smooth systems, is as complicated as the general isomorphism problem for ergodic measure-preserving systems. The authors prove that the isomorphism problem is impossible even for diffeomorphisms of compact surfaces. It is pointed out that the problem of finding even one measure-preserving transformation not isomorphic to its inverse is difficult.
Finally the authors state two still open problems.
Reviewer: Ion Mihai (Bucureşti)On the complexity of fitted toral dynamicshttps://zbmath.org/1517.370322023-09-22T14:21:46.120933Z"Maller, Michael"https://zbmath.org/authors/?q=ai:maller.michael"Whitehead, Jennifer"https://zbmath.org/authors/?q=ai:whitehead.jenniferSummary: In earlier work we defined a computational saddle transition problem which arises in the dynamics of certain hyperbolic toral automorphisms, and proved, using the shadowing lemma, that in an appropriate model of computation this problem is in Oracle \(\mathbf{NP}\), up to a highly restricted oracle. In this note we show similar methods can be extended to a far larger class of dynamical systems, a class which is dense in the \(C^0\)-topology on Diff\(^1(\mathbf{T}^{\mathbf{2}})\). We adapt the fitted diffeomorphisms of \textit{M. Shub} and \textit{D. Sullivan} [Topology 14, 109--132 (1975; Zbl 0408.58023)]
on the 2-Torus to a computational framework. Just as in their case, the resulting ``well-fitted'' toral automorphisms are structurally stable, and \(C^0\)-dense, and we show the associated saddle transition problems are, in our model, in Oracle \(\mathbf{NP}\).A dynamic Parrondo's paradox for continuous seasonal systemshttps://zbmath.org/1517.370332023-09-22T14:21:46.120933Z"Cima, Anna"https://zbmath.org/authors/?q=ai:cima.anna"Gasull, Armengol"https://zbmath.org/authors/?q=ai:gasull.armengol"Mañosa, Víctor"https://zbmath.org/authors/?q=ai:manosa.victor(no abstract)On the topological structure of manifolds supporting axiom a systemshttps://zbmath.org/1517.370342023-09-22T14:21:46.120933Z"Grines, Vyacheslav Z."https://zbmath.org/authors/?q=ai:grines.vyacheslav-z"Medvedev, Vladislav S."https://zbmath.org/authors/?q=ai:medvedev.vladislav-s"Zhuzhoma, Evgeny V."https://zbmath.org/authors/?q=ai:zhuzhoma.evgenii-vSummary: Let \(M^n\), \(n\geqslant 3\), be a closed orientable \(n\)-manifold and \(\mathbb{G}(M^n)\) the set of A-diffeomorphisms \(f:M^n\to M^n\) whose nonwandering set satisfies the following conditions: \((1)\) each nontrivial basic set of the nonwandering set is either an orientable codimension one expanding attractor or an orientable codimension one contracting repeller; \((2)\) the invariant manifolds of isolated saddle periodic points intersect transversally and codimension one separatrices of such points can intersect only one-dimensional separatrices of other isolated periodic orbits. We prove that the ambient manifold \(M^n\) is homeomorphic to either the sphere \(\mathbb{S}^n\) or the connected sum of \(k_f\geqslant 0\) copies of the torus \(\mathbb{T}^n\), \(\eta_f\geqslant 0\) copies of \(\mathbb{S}^{n-1}\times\mathbb{S}^1\) and \(l_f\geqslant 0\) simply connected manifolds \(N^n_1,\dots,N^n_{l_f}\) which are not homeomorphic to the sphere. Here \(k_f\geqslant 0\) is the number of connected components of all nontrivial basic sets, \(\eta_f=\frac{\kappa_f}{2}-k_f+\frac{\nu_f-\mu_f+2}{2},\kappa_f\geqslant 0\) is the number of bunches of all nontrivial basic sets, \(\mu_f\geqslant 0\) is the number of sinks and sources, \(\nu_f\geqslant 0\) is the number of isolated saddle periodic points with Morse index \(1\) or \(n-1\), \(0\leqslant l_f\leqslant\lambda_f\), \(\lambda_f\geqslant 0\) is the number of all periodic points whose Morse index does not belong to the set \(\{0,1,n-1,n\}\) of diffeomorphism \(f\). Similar statements hold for gradient-like flows on \(M^n\). In this case there are no nontrivial basic sets in the nonwandering set of a flow. As an application, we get sufficient conditions for the existence of heteroclinic intersections and periodic trajectories for Morse-Smale flows.Decay of correlations for certain isometric extensions of Anosov flowshttps://zbmath.org/1517.370352023-09-22T14:21:46.120933Z"Siddiqi, Salman"https://zbmath.org/authors/?q=ai:siddiqi.salmanSummary: We establish exponential decay of correlations of all orders for locally \(G\)-accessible isometric extensions of transitive Anosov flows, under the assumption that the strong stable and strong unstable distributions of the base Anosov flow are \(C^1\). This is accomplished by translating accessibility properties of the extension into local non-integrability estimates measured by infinitesimal transitivity groups used by \textit{D. Dolgopyat} [Isr. J. Math. 130, 157--205 (2002; Zbl 1005.37005)], from which we obtain contraction properties for a class of `twisted' symbolic transfer operators.Entropy of irregular points for some dynamical systemshttps://zbmath.org/1517.370362023-09-22T14:21:46.120933Z"Gelfert, Katrin"https://zbmath.org/authors/?q=ai:gelfert.katrin"Pacifico, Maria José"https://zbmath.org/authors/?q=ai:pacifico.maria-jose"Sanhueza, Diego"https://zbmath.org/authors/?q=ai:sanhueza.diegoSummary: We derive sufficient conditions for a dynamical systems to have a set of irregular points with full topological entropy. Such conditions are verified for nonuniformly hyperbolic systems such as positive entropy surface diffeomorphisms and rational functions on the Riemann sphere.A multiplicative ergodic theoretic characterization of relative equilibrium stateshttps://zbmath.org/1517.370372023-09-22T14:21:46.120933Z"Antonioli, John"https://zbmath.org/authors/?q=ai:antonioli.john"Hong, Soonjo"https://zbmath.org/authors/?q=ai:hong.soonjo"Quas, Anthony"https://zbmath.org/authors/?q=ai:quas.anthony-nSummary: In this article, we continue the structural study of factor maps between symbolic dynamical systems and the relative thermodynamic formalism. Here, one is studying a factor map from a shift of finite type \(X\) (equipped with a potential function) to a sofic shift \(Z\), equipped with a shift-invariant measure \(\nu\). We study relative equilibrium states, that is, shift-invariant measures on \(X\) that push forward under the factor map to \(\nu\) which maximize the relative pressure: the relative entropy plus the integral of \(\phi\). In this paper, we establish a new connection to multiplicative ergodic theory by relating these factor triples to a cocycle of Ruelle-Perron-Frobenius operators, and showing that the principal Lyapunov exponent of this cocycle is the relative pressure; and the dimension of the leading Oseledets space is equal to the number of measures of relative maximal entropy, counted with a previously identified concept of multiplicity.On relationships between the spectral potential of transfer operators, \(t\)-entropy, entropy and topological pressurehttps://zbmath.org/1517.370382023-09-22T14:21:46.120933Z"Bakhtin, V. I."https://zbmath.org/authors/?q=ai:bakhtin.victor-i"Lebedev, A. V."https://zbmath.org/authors/?q=ai:lebedev.andrei-vladimirovichSummary: The paper is devoted to the analysis of relationships between principal objects of the spectral theory of dynamical systems (transfer and weighted shift operators) and basic characteristics of information theory and thermodynamic formalism (entropy and topological pressure). We present explicit formulas linking these objects with the \(t\)-entropy and spectral potential. Herewith we uncover the role of inverse rami-rate, the forward entropy along with an essential set, and the property of noncontractibility of a dynamical system.New approach to weighted topological entropy and pressurehttps://zbmath.org/1517.370392023-09-22T14:21:46.120933Z"Tsukamoto, Masaki"https://zbmath.org/authors/?q=ai:tsukamoto.masakiSummary: Motivated by fractal geometry of self-affine carpets and sponges, \textit{D.-J. Feng} and \textit{W. Huang} [J. Math. Pures Appl. (9) 106, No. 3, 411--452 (2016; Zbl 1360.37080)] introduced weighted topological entropy and pressure for factor maps between dynamical systems, and proved variational principles for them. We introduce a new approach to this theory. Our new definitions of weighted topological entropy and pressure are very different from the original definitions of \textit{D.-J. Feng} and \textit{W. Huang} [loc. cit.]. The equivalence of the two definitions seems highly non-trivial. Their equivalence can be seen as a generalization of the dimension formula for the Bedford-McMullen carpet in purely topological terms.Distribution in the unit tangent bundle of the geodesics of given typehttps://zbmath.org/1517.370402023-09-22T14:21:46.120933Z"Erlandsson, Viveka"https://zbmath.org/authors/?q=ai:erlandsson.viveka"Souto, Juan"https://zbmath.org/authors/?q=ai:souto.juanSummary: Recall that two geodesics in a negatively curved surface \(S\) are of the same type if their free homotopy classes differ by a homeomorphism of the surface. In this note we study the distribution in the unit tangent bundle of the geodesics of fixed type, proving that they are asymptotically equidistributed with respect to a certain measure \(\mathfrak{m}^S\) on \(T^1S\). We study a few properties of this measure, showing for example that it distinguishes between hyperbolic surfaces.Typical properties of periodic Teichmüller geodesics: Lyapunov exponentshttps://zbmath.org/1517.370412023-09-22T14:21:46.120933Z"Hamenstädt, Ursula"https://zbmath.org/authors/?q=ai:hamenstadt.ursulaSummary: Consider a component \(\mathcal{Q}\) of a stratum in the moduli space of area-one abelian differentials on a surface of genus \(g\). Call a property \(\mathcal{P}\) for periodic orbits of the Teichmüller flow on \(\mathcal{Q}\) \textit{typical} if the growth rate of orbits with property \(\mathcal{P}\) is maximal. We show that the following property is typical. Given a continuous integrable cocycle over the Teichmüller flow with values in a vector bundle \(V\to\mathcal{Q}\), the logarithms of the eigenvalues of the matrix defined by the cocycle and the orbit are arbitrarily close to the Lyapunov exponents of the cocycle for the Masur-Veech measure.Measure rigidity of Anosov flows via the factorization methodhttps://zbmath.org/1517.370422023-09-22T14:21:46.120933Z"Katz, Asaf"https://zbmath.org/authors/?q=ai:katz.asafSummary: Using the factorization method of \textit{A. Eskin} and \textit{M. Mirzakhani} [Publ. Math., Inst. Hautes Étud. Sci. 127, 95--324 (2018; Zbl 1478.37002)], we show that generalized \(u\)-Gibbs states over quantitatively non-integrable partially hyperbolic systems have absolutely continuous disintegrations on unstable manifolds. As an application, we show a pointwise equidistribution theorem analogous to the equidistribution results of \textit{D. Kleinbock} et al. [Math. Ann. 367, No. 1--2, 857--879 (2017; Zbl 1417.37056)] and \textit{A. Eskin} and \textit{J. Chaika} [J. Mod. Dyn. 9, 1--23 (2015; Zbl 1358.37008)].Isolations of geodesic planes in the frame bundle of a hyperbolic 3-manifoldhttps://zbmath.org/1517.370432023-09-22T14:21:46.120933Z"Mohammadi, Amir"https://zbmath.org/authors/?q=ai:mohammadi.amir-amjad|mohammadi.amir|mohammadi.amir-hossein-mousavi"Oh, Hee"https://zbmath.org/authors/?q=ai:oh.hee-keun|oh.heeSummary: We present a quantitative isolation property of the lifts of properly immersed geodesic planes in the frame bundle of a geometrically finite hyperbolic \(3\)-manifold. Our estimates are polynomials in the tight areas and Bowen-Margulis-Sullivan densities of geodesic planes, with degree given by the modified critical exponents.A polynomial chaos expansion approach for nonlinear dynamic systems with interval uncertaintyhttps://zbmath.org/1517.370442023-09-22T14:21:46.120933Z"Wang, Liqun"https://zbmath.org/authors/?q=ai:wang.liqun"Chen, Zengtao"https://zbmath.org/authors/?q=ai:chen.zengtao"Yang, Guolai"https://zbmath.org/authors/?q=ai:yang.guolai(no abstract)On the topological entropy of \((a,b)\)-continued fraction transformationshttps://zbmath.org/1517.370452023-09-22T14:21:46.120933Z"Abrams, Adam"https://zbmath.org/authors/?q=ai:abrams.adam"Katok, Svetlana"https://zbmath.org/authors/?q=ai:katok.svetlana-r"Ugarcovici, Ilie"https://zbmath.org/authors/?q=ai:ugarcovici.ilieSummary: We study the topological entropy of a two-parameter family of maps related to \((a,b)\)-continued fraction algorithms and prove that it is constant on a square within the parameter space (two vertices of this square correspond to well-studied continued fraction algorithms). The proof uses conjugation to maps of constant slope. We also present experimental evidence that the topological entropy is flexible (i.e. takes any value in a range) on the whole parameter space.Invariant measures for interval maps without Lyapunov exponentshttps://zbmath.org/1517.370462023-09-22T14:21:46.120933Z"Olivares-Vinales, Jorge"https://zbmath.org/authors/?q=ai:olivares-vinales.jorgeSummary: We construct an invariant measure for a piecewise analytic interval map whose Lyapunov exponent is not defined. Moreover, for a set of full measure, the pointwise Lyapunov exponent is not defined. This map has a Lorenz-like singularity and non-flat critical points.Crinkled changes of variables for maps on a circlehttps://zbmath.org/1517.370472023-09-22T14:21:46.120933Z"Das, Suddhasattwa"https://zbmath.org/authors/?q=ai:das.suddhasattwa"Yorke, James A."https://zbmath.org/authors/?q=ai:yorke.james-a(no abstract)Existence of the FS-type renormalisation fixed point for unidirectionally-coupled pairs of mapshttps://zbmath.org/1517.370482023-09-22T14:21:46.120933Z"Burbanks, Andrew"https://zbmath.org/authors/?q=ai:burbanks.andrew-d"Osbaldestin, Andrew"https://zbmath.org/authors/?q=ai:osbaldestin.andrew-hSummary: We give the first proof of the existence of a renormalisation fixed-point for period-doubling in pairs of maps of two variables lying in the so-called Feigenbaum-Summation (FS) universality class. The first map represents a subsystem that is unimodal with an extremum of degree two. The dynamics of the second map accumulates an integral characteristic of the dynamics of the first, via a particular form of unidirectional coupling. We prove the existence of the corresponding renormalisation fixed point by rigorous computer-assisted means and gain tight rigorous bounds on the associated universal constants. Our work provides the first step in establishing rigorously the picture conjectured by \textit{S. P. Kuznetsov} et al. [J. Stat. Phys. 130, No. 3, 599--616 (2008; Zbl 1140.82023)]
of the birth, from the FS-type fixed point, of the so-called C-type two-cycle via a period doubling in the dynamics of the renormalisation group transformation itself.Asymptotic scaling and universality for skew products with factors in \(\mathrm{SL}(2,\mathbb{R})\)https://zbmath.org/1517.370492023-09-22T14:21:46.120933Z"Koch, Hans"https://zbmath.org/authors/?q=ai:koch.hans-friedrichSummary: We consider skew-product maps over circle rotations \(x\mapsto x+\alpha\pmod 1\) with factors that take values in \(\mathrm{SL}(2,\mathbb{R})\). In numerical experiments, with \(\alpha\) the inverse golden mean, Fibonacci iterates of maps from the almost Mathieu family exhibit asymptotic scaling behavior that is reminiscent of critical phase transitions. In a restricted setup that is characterized by a symmetry, we prove that critical behavior indeed occurs and is universal in an open neighborhood of the almost Mathieu family. This behavior is governed by a periodic orbit of a renormalization transformation. An extension of this transformation is shown to have a second periodic orbit as well, and we present some evidence that this orbit attracts supercritical almost Mathieu maps.Attractor separation and signed cycles in asynchronous Boolean networkshttps://zbmath.org/1517.370502023-09-22T14:21:46.120933Z"Richard, Adrien"https://zbmath.org/authors/?q=ai:richard.adrien"Tonello, Elisa"https://zbmath.org/authors/?q=ai:tonello.elisaSummary: The structure of the graph defined by the interactions in a Boolean network can determine properties of the asymptotic dynamics. For instance, considering the asynchronous dynamics, the absence of positive cycles guarantees the existence of a unique attractor, and the absence of negative cycles ensures that all attractors are fixed points. In presence of multiple attractors, one might be interested in properties that ensure that attractors are sufficiently ``isolated'', that is, they can be found in separate subspaces or even trap spaces, subspaces that are closed with respect to the dynamics. Here we introduce notions of separability for attractors and identify corresponding necessary conditions on the interaction graph. In particular, we show that if the interaction graph has at most one positive cycle, or at most one negative cycle, or if no positive cycle intersects a negative cycle, then the attractors can be separated by subspaces. If the interaction graph has no path from a negative to a positive cycle, then the attractors can be separated by trap spaces. Furthermore, we study networks with interaction graphs admitting two vertices that intersect all cycles, and show that if their attractors cannot be separated by subspaces, then their interaction graph must contain a copy of the complete signed digraph on two vertices, deprived of a negative loop. We thus establish a connection between a dynamical property and a complex network motif. The topic is far from exhausted and we conclude by stating some open questions.Antisymmetric diffeomorphisms and bifurcations of a double conservative Hénon maphttps://zbmath.org/1517.370512023-09-22T14:21:46.120933Z"Gonchenko, Sergey V."https://zbmath.org/authors/?q=ai:gonchenko.sergey-v"Safonov, Klim A."https://zbmath.org/authors/?q=ai:safonov.klim-a"Zelentsov, Nikita G."https://zbmath.org/authors/?q=ai:zelentsov.nikita-gSummary: We propose a new method for constructing multidimensional reversible maps by only two input data: a diffeomorphism \(T_1\) and an involution \(h\), i.e., a map (diffeomorphism) such that \(h^2=Id\). We construct the desired reversible map \(T\) in the form \(T=T_1\circ T_2\), where \(T_2=h\circ T_1^{-1}\circ h\). We also discuss how this method can be used to construct normal forms of Poincaré maps near mutually symmetric pairs of orbits of homoclinic or heteroclinic tangencies in reversible maps. One of such normal forms, as we show, is a two-dimensional double conservative Hénon map \(H\) of the form \(\bar{x}=M+cx-y^2\); \(y=M+c\bar{y}-\bar{x}^2 \). We construct this map by the proposed method for the case when \(T_1\) is the standard Hénon map and the involution \(h\) is \(h:(x,y)\to(y,x)\). For the map \(H\), we study bifurcations of fixed and period-2 points, among which there are both standard bifurcations (parabolic, period-doubling and pitchfork) and singular ones (during transition through \(c=0)\).Unbounded fast escaping wandering domainshttps://zbmath.org/1517.370522023-09-22T14:21:46.120933Z"Evdoridou, Vasiliki"https://zbmath.org/authors/?q=ai:evdoridou.vasiliki"Glücksam, Adi"https://zbmath.org/authors/?q=ai:glucksam.adi"Pardo-Simón, Leticia"https://zbmath.org/authors/?q=ai:pardo-simon.leticiaThis paper is focused on wandering domains of transcendental entire functions. By using approximation theory, the authors construct:
\begin{itemize}
\item Transcendental entire functions with an orbit of unbounded fast escaping wandering domains, answering a question of \textit{P. J. Rippon} and \textit{G. M. Stallard} [Proc. Lond. Math. Soc. (3) 105, No. 4, 787--820 (2012; Zbl 1291.30160)];
\item Transcendental entire functions with an unbounded fast escaping wandering domain and with any prescribed order of growth in \((1/2, 1]\), which is a weaker version of a conjecture of \textit{I. N. Baker} [J. Aust. Math. Soc., Ser. A 30, 483--495 (1981; Zbl 0474.30023)];
\item Transcendental entire functions with unbounded fast escaping connected wandering domains which realize all possible dynamical behaviors in terms of convergence to the boundary.
\end{itemize}
Reviewer: Weiwei Cui (Lund)Single and double toral band Fatou components in meromorphic dynamicshttps://zbmath.org/1517.370532023-09-22T14:21:46.120933Z"Hawkins, Jane"https://zbmath.org/authors/?q=ai:hawkins.jane-m"Koss, Lorelei"https://zbmath.org/authors/?q=ai:koss.loreleiThe authors study the dynamics under iteration of elliptic functions (i.e., doubly periodic meromorphic functions) in unbounded Fatou components. They call these domains toral bands (Definition 1.1). More precisely, for an elliptic function \(f_{\Lambda}\), a Fatou component is a toral band if it contains an open simply connected set which projects to a topological band around the torus \(\mathbb{C}/\Lambda\) containing a curve which is not homotopically trivial.
The main results of the paper show that toral bands cannot contain Siegel disks and Herman rings, but can contain parabolic cycles. In general, toral bands need not be periodic. Moreover, in the absence of Herman rings, every toral band contains at least two critical points. Some examples are also given, hence illustrating that the above properties are essentially the only restrictions for toral bands.
Reviewer: Weiwei Cui (Lund)Dynamics of generalised exponential mapshttps://zbmath.org/1517.370542023-09-22T14:21:46.120933Z"Comdühr, Patrick"https://zbmath.org/authors/?q=ai:comduhr.patrick"Evdoridou, Vasiliki"https://zbmath.org/authors/?q=ai:evdoridou.vasiliki"Sixsmith, David J."https://zbmath.org/authors/?q=ai:sixsmith.david-jComplex dynamics studies dynamical systems generated by holomorphic or meromorphic functions (in one or several variables). This field has attracted a lot of attention, in particular, since the 80s of last century. One direction of extension is to study dynamics generated by quasiregular maps. The present paper belongs to this research trend, and it is restricted to exponential functions.
Dynamical properties of exponential functions have been studied quite a lot. It is well known now that if an exponential function is of disjoint type, then the Julia set is a Cantor bouquet. This applies to functions of the form \(e^z -a\), where \(a>1\). Here the authors study ``non-analytic'' version of such functions, which they call generalised exponentials (Definition 1.1), and prove that the set \(J\) of points which are not attracted by the attracting fixed point has the structure of Cantor bouquet. Some other properties concerning \(J\) are also proved (Theorem 1.4).
Reviewer: Weiwei Cui (Lund)On dynamical gaskets generated by rational maps, Kleinian groups, and Schwarz reflectionshttps://zbmath.org/1517.370552023-09-22T14:21:46.120933Z"Lodge, Russell"https://zbmath.org/authors/?q=ai:lodge.russell"Lyubich, Mikhail"https://zbmath.org/authors/?q=ai:lyubich.mikhail"Merenkov, Sergei"https://zbmath.org/authors/?q=ai:merenkov.sergei"Mukherjee, Sabyasachi"https://zbmath.org/authors/?q=ai:mukherjee.sabyasachiThe authors add a new contribution to the so-called \textit{Fatou-Sullivan dictionary}, which connects two branches of conformal dynamics -- iterations of rational maps and actions of discrete subgroups of Möbius transformations (i.e., Kleinian groups) on \(\hat{\mathbb{C}}\).
In particular, they focus on a family of fractals, named \textit{gaskets}, coming from circle packings and triangulations of the sphere. Given a triangulation \(\mathcal{T}\) of the sphere \(S^2\cong\hat{\mathbb{C}}\), the Circle Packing Theorem of Koebe-Andreev-Thurston (see [\textit{W. P. Thurston}, The geometry and topology of three-manifolds. Providence, RI: American Mathematical Society (AMS) (2022; Zbl 1507.57005)]) guarantees the existence of a circle packing \(\mathcal{C}\) whose \textit{nerve} (a graph encoding the tangency data of the circles) is \(\mathcal{T}\), up to isotopy. Each face \(f\) of \(\mathcal{T}\) gives three mutually tangent circles in \(\mathcal{C}\), and let \(R_f\) be the reflection in the unique circle passing through the three points of tangency. Then the group \(H_\mathcal{T}\) generated by \(\{R_f\}_{f\text{ a face of } \mathcal{T}}\) acts on \(\hat{\mathbb{C}}\) as conformal and anticonformal maps. The limit set of \(H_\mathcal{T}\) is called a \textit{round gasket}. Notice that the complement of a round gasket is a disjoint union of round disks, including the disks bounded by circles in \(\mathcal{C}\). A \textit{gasket} is any subset of \(\hat{\mathbb{C}}\) homeomorphic to a round gasket. The classical Apollonian gasket is an example of a round gasket, associated to the tetrahedral triangulation of the sphere.
By definition, gaskets arise as limit sets of Kleinian groups. They also appear as Julia sets of (anti)rational maps. As a matter of fact, in the paper, for each triangulation \(\mathcal{T}\) of the sphere, the authors construct an antirational map \(g\) whose Julia set is homeomorphic to the limit set of \(H_\mathcal{T}\), by applying Thurston's topological characterization of rational maps [\textit{A. Douady} and \textit{J. H. Hubbard}, Acta Math. 171, No. 2, 263--297 (1993; Zbl 0806.30027)]. In the case of the Apollonian gasket, they also construct an explicit anti-quasiregular model for the map \(g\).
The connection between the group \(H\) and the map \(g\) is dynamical. In fact, the authors show that they can be mated via David surgery and Schwartz reflection to produce a hybrid dynamical system.
One motivation of the paper is to study the group of quasisymmetric homeomorphisms of a fractal. Previously, this group was determined for a class of Sierpiński carpet Julia sets [\textit{M. Bonk} et al., Adv. Math. 301, 383--422 (2016; Zbl 1358.37083)] and Basilica [\textit{M. Lyubich} and \textit{S. Merenkov}, Geom. Funct. Anal. 28, No. 3, 727--754 (2018; Zbl 1499.30238)]. Here the authors consider this problem for a large family of dynamical gaskets (limit sets and Julia sets).
More specifically, under a weak assumption on \(\mathcal{T}\), they show the following:
\begin{itemize}
\item For the limit set \(\Lambda_H\) of \(H\), the group of homeomorphisms of \(\Lambda_H\) is the semidirect product of the (finite) group of symmetries of \(\mathcal{T}\) with \(H\). In particular, every homeomorphism is conformal or anticonformal;
\item For the Julia set \(\mathcal{J}_g\) of \(g\), the group of homeomorphisms coincide with the group of quasisymmetries;
\item There exists a dynamically defined homeomorphism \(h:\Lambda_H\to\mathcal{J}_g\), so that it induces an isomorphism between the groups of homeomorphisms (and hence quasisymmetries).
\end{itemize}
The homeomorphism \(h\) mentioned above is \emph{not} quasiconformal, so the fact that it induces an isomorphism between groups of quasisymmetries is not automatic. This also means that the group of quasisymmetries does not distinguish the quasisymmetric type of \(\Lambda_H\) and \(\mathcal{J}_g\).
Reviewer: Yongquan Zhang (Stony Brook)A non-singular version of the Oseledeč ergodic theoremhttps://zbmath.org/1517.370562023-09-22T14:21:46.120933Z"Dooley, Anthony H."https://zbmath.org/authors/?q=ai:dooley.anthony-haynes"Jin, Jie"https://zbmath.org/authors/?q=ai:jin.jieSummary: Kingman's subadditive ergodic theorem is traditionally proved in the setting of a measure-preserving invertible transformation \(T\) of a measure space \((X,\mu)\). We use a theorem of \textit{C. E. Silva} and \textit{P. Thieullen} [J. Math. Anal. Appl. 154, No. 1, 83--99 (1991; Zbl 0719.28008)]
to extend the theorem to the setting of a not necessarily invertible transformation, which is non-singular under the assumption that \(\mu\) and \(\mu\circ T\) have the same null sets. Using this, we are able to produce versions of the Furstenberg-Kesten theorem and the Oseledeč ergodic theorem for products of random matrices without the assumption that the transformation is either invertible or measure-preserving.Statistical stability and linear response for random hyperbolic dynamicshttps://zbmath.org/1517.370572023-09-22T14:21:46.120933Z"Dragičević, Davor"https://zbmath.org/authors/?q=ai:dragicevic.davor"Sedro, Julien"https://zbmath.org/authors/?q=ai:sedro.julienSummary: We consider families of random products of close-by Anosov diffeomorphisms, and show that statistical stability and linear response hold for the associated families of equivariant and stationary measures. Our analysis relies on the study of the top Oseledets space of a parametrized transfer operator cocycle, as well as ad-hoc abstract perturbation statements. As an application, we show that, when the quenched central limit theorem (CLT) holds, under the conditions that ensure linear response for our cocycle, the variance in the CLT depends differentiably on the parameter.Random Young towers and quenched limit lawshttps://zbmath.org/1517.370582023-09-22T14:21:46.120933Z"Su, Yaofeng"https://zbmath.org/authors/?q=ai:su.yaofengSummary: We obtain quenched almost sure invariance principles (with convergence rates) for random Young towers if the average measure of the tail of return times to the base of random towers decays sufficiently fast. We apply our results to some independent and identically distributed perturbations of some non-uniformly expanding maps. These imply that the random systems under study tend to a Brownian motion under various scalings.Non-Hamiltonian actions with fewer isolated fixed pointshttps://zbmath.org/1517.370592023-09-22T14:21:46.120933Z"Jang, Donghoon"https://zbmath.org/authors/?q=ai:jang.donghoon"Tolman, Susan"https://zbmath.org/authors/?q=ai:tolman.susanThe following question is known as ``McDuff conjecture'': Does there exist a non-Hamiltonian symplectic circle action on a closed, connected symplectic manifold with a non-empty discrete fixed set? \textit{S. Tolman} [Invent. Math. 210, No. 3, 877--910 (2017; Zbl 1383.53061)] answered this question by constructing a non-Hamiltonian symplectic circle action on a 6-dimensional closed, connected symplectic manifold with exactly \(32\) isolated fixed points.
In this paper the authors improve this example.
More concretely, they obtain the following main theorem:
Theorem. Given any integer \(k \geqslant 5\), there exists a non-Hamiltonian symplectic circle action on a closed, connected 6-dimensional symplectic manifold with exactly \(2k\) fixed points.
From the proof of the above theorem there stems another result:
Theorem. There exists a non-Hamiltonian symplectic circle action on a closed, connected 6-dimensional symplectic manifold \((M, \omega)\) with a generalized moment map \(\Psi: M \to \mathbb{R}/10\mathbb{Z}\simeq S^1\) satisfying the following properties: each of the level sets \(\Psi^{-1}(\pm 1)\) contains \(5\) fixed points with weights \(\pm \{2,-1,-1\}\); otherwise, the action is locally free.
The Duistermaat-Heckman function is
\[
\varphi(t)=\begin{cases} 12 -2t^2, & -1 \leqslant t \leqslant 1,\\
2 + \frac{1}{2}( t -5 )^2, & 1\leqslant t \leqslant 9. \end{cases}
\]
The reduced space \(M/_{t} S^1\) is diffeomorphic to a generalized K3 surface with \(5\) isolated \(\mathbb{Z}_2\) singularities for all \(t\in (1, 9)\), and symplectomorphic to a tame K3 surface for all \(t\in (-1, 1)\).
Finally, the authors show that any non-Hamiltonian symplectic circle action on a closed, connected 6-dimensional symplectic manifold must have at least \(10\) fixed points if it shares a number of key properties with the examples constructed in the last theorem. Indeed, one has:
Lemma. Let the circle act on a closed connected 6-dimensional symplectic manifold \((M, \omega)\) with generalized moment map \(\Psi: M\to \mathbb{R}/\mathbb{Z}\). Assume that the \(S^1\) action on \(\Psi^{-1}(0)\) is free, that the reduced space \(M//_{0} S^1\) is a K3 surface, and that every fixed point \(p\in M^{S^1}\) has weights \(\pm \{2,-1,-1\}\). Then \(M\) has at least \(10\) fixed points.
Reviewer: Jianbo Wang (Tianjin)Almost differentially nondegenerate Nijenhuis operatorshttps://zbmath.org/1517.370602023-09-22T14:21:46.120933Z"Akpan, D. Zh."https://zbmath.org/authors/?q=ai:akpan.d-zhSummary: The paper is devoted to the study of Nijenhuis operators of arbitrary dimension \(n\) in a neighborhood of a point at which the first \(n-1\) coefficients of the characteristic polynomial are functionally independent and the last coefficient (the determinant of the operator) is an arbitrary function. We prove a theorem on the general form of such Nijenhuis operators and also obtain their complete description for the case in which the determinant has a nondegenerate singularity.Flexibility of sections of nearly integrable Hamiltonian systemshttps://zbmath.org/1517.370612023-09-22T14:21:46.120933Z"Burago, Dmitri"https://zbmath.org/authors/?q=ai:burago.dmitri"Chen, Dong"https://zbmath.org/authors/?q=ai:chen.dong"Ivanov, Sergei"https://zbmath.org/authors/?q=ai:ivanov.sergei-vladimirovichThe authors begin with a symplectic transformation of the ball \(D^{2n}\) with \(n \geq 2\) that is close to the identity in the \(C^\infty\) norm and a completely integrable Hamiltonian system in \(\mathbb{R}^{2n}\). They show that there exist \(C^\infty\) perturbations, arbitrarily small in the \(C^\infty\) norm, so that each resulting perturbed flow contains a natural Poincaré section. On this section -- after proper rescaling and iterations -- the return map is precisely the given transformation. Further, the section occurs near some periodic orbit of the unperturbed flow.
The proofs of the authors' results make use of an extended version of the ``dual lens map'' technique described in [\textit{D. Burago} and \textit{S. Ivanov}, Geom. Topol. 20, No. 1, 469--490 (2016; Zbl 1350.53094)]. In the current paper the technique is applied to Lagrangian submanifolds in a symplectic manifold instead of geodesics on a Finsler manifold as in the cited reference.
Reviewer: William J. Satzer Jr. (St. Paul)Persistence of degenerate hyperbolic lower-dimensional invariant tori in Hamiltonian systems with Bruno's conditionshttps://zbmath.org/1517.370622023-09-22T14:21:46.120933Z"Yang, Xiaomei"https://zbmath.org/authors/?q=ai:yang.xiaomei"Xu, Junxiang"https://zbmath.org/authors/?q=ai:xu.junxiangThe authors investigate Hamiltonian systems with Bruno non-degeneracy conditions that give rise to persistent degenerate lower dimensional tori. This paper proves a generalizion of the result in [\textit{J. Xu} and \textit{J. You}, Regul. Chaotic Dyn. 25, No. 6, 616--650 (2020; Zbl 1481.37067)]. They consider the Hamiltonian system \[H = \langle \omega_0, y\rangle + \frac{1}{2} \langle My, y \rangle + \langle \tilde{a}, y \rangle u + \frac{1}{2} a u^2 - \frac{1}{2} b v^2 + u^{2d} + P,\] where \((x,y,u,v) \in \mathbb{T}^n \times \mathbb{R}^n \times \mathbb{R} \times \mathbb{R}\), having symplectic structure \(dx \wedge dy + du \wedge dv\). \(M\) is a symmetric matrix, \(\omega_0\) is a Diophantine vector, and \(a\), \(b\), \(d\) and \(\tilde{a}\) are constants, and \(P\) represents a perturbation.
The main result is that, under certain conditions, the Hamiltonian \(H\) admits an invariant torus with a frequency vector that is a small dilation of \(\omega_0\). These conditions include: a requirement that the perturbation \(P\) is small, a Diophantine condition on \(\omega_0\), a hyperbolicity condition, the Bruno non-degeneracy condition that det\((M) = 0\) and rank\((\omega_0, M) = n\), as well as a degeneracy condition that implies the hyperbolic torus \(\mathbb{T}^n \times \{0, 0, 0 \}\) is degenerate.
The proof is similar to that in [loc. cit.], except that with the Bruno non-degeneracy condition, the authors can control \(n-1\) components of frequencies so that the frequency vector can always be a small dilation of the prescribed one.
Reviewer: William J. Satzer Jr. (St. Paul)Coexistence of hyperbolic and elliptic invariant tori for completely degenerate quasi-periodically forced mapshttps://zbmath.org/1517.370632023-09-22T14:21:46.120933Z"Zhou, Guangzhao"https://zbmath.org/authors/?q=ai:zhou.guangzhao"Zhang, Yuan"https://zbmath.org/authors/?q=ai:zhang.yuan"Si, Wen"https://zbmath.org/authors/?q=ai:si.wenSummary: Consider the following completely degenerate quasi-periodically forced skew-product maps of the form
\[
\begin{cases}
\bar{x}& = x+y^m+\epsilon f_1(x,y,\theta,\epsilon)+h_1(x,y,\theta,\epsilon),\\
\bar{y}& = y+ \lambda x^n+\epsilon f_2(x,y,\theta,\epsilon)+h_2(x,y,\theta,\epsilon),\\
\bar{\theta}& = \theta+\omega,
\end{cases}
\]
where \((x,y,\theta)\in\mathbb{R}\times\mathbb{R}\times\mathbb{T}^d\), \(\lambda=\pm 1\), \(\omega\in\mathbb{R}^d\), \(n\) and \(m\) are positive integers satisfying \(n\geq m\), \(mn>1\), \(f_1\), \(f_2\), \(h_1\), \(h_2\) are real analytic on \((x,y,\theta)\) and \(C^1\)-Whitney smooth on \(\epsilon\), and \(h_1, h_2=\mathcal{O}(|(x,y)|^{n+1})\). The existence of weak-hyperbolic (weak-elliptic) invariant tori for the above maps with \(\lambda=1\) (\(\lambda = -1\)) has been proved in [\textit{W. Si} and \textit{Y. Yi}, Nonlinearity 33, No. 11, 6072--6098 (2020; Zbl 1455.37048); \textit{T. Zhang} et al., Discrete Contin. Dyn. Syst. 36, No. 11, 6599--6622 (2016; Zbl 1376.37108)]. In this paper, we prove the above completely degenerate skew-product map both in the case \(\lambda=1\) and in the case \(\lambda = -1\) not only admits weak-hyperbolic invariant tori but also weak-elliptic invariant tori under the suitable conditions. Moreover, the number of invariant tori is investigated in both cases. See Theorem 2.1 and Theorem 2.2. The results of this paper are the situations that are not discussed in the existing literature.Brake orbits with minimal period estimates of first-order variant subquadratic Hamiltonian systemshttps://zbmath.org/1517.370642023-09-22T14:21:46.120933Z"Zhang, Xiaofei"https://zbmath.org/authors/?q=ai:zhang.xiaofei"Wang, Fanjing"https://zbmath.org/authors/?q=ai:wang.fanjingThe authors investigate the existence of \(\tau\)-periodic brake orbits (\(\tau > 0\)) of the autonomous first-order Hamiltonian system
\[
\left \{ \begin{array}{l} J\dot{z}(t)=-\nabla H(z(t)), \\
z(-t)=Nz(t), \\
z(t+\tau )=z(t), \end{array} \right.
\]
where \(t\in \mathbb R, H\in C^2(\mathbb R^{2n},\mathbb R)\) with \(H(Nz)=H(z), z\in \mathbb R^{2n}\), \[N= \left( \begin{array}{l l} -I_n & 0 \\
0 & I_n \end{array} \right),\]
\[J= \left( \begin{array}{l l} 0 & -I_n \\
I_n & 0 \end{array} \right). \]
Under a generalized subquadratic growth condition, existence of brake orbits are guaranteed via the homological link theorem in [\textit{A. Abbondandolo}, Morse theory for Hamiltonian systems. Boca Raton, FL: Chapman \& Hall/CRC (2001; Zbl 0967.37002)]. The minimal period estimate is given by the Morse index and \(L_0\)-index estimates (see [\textit{Y. Wang}, et al., Acta Autom. Sin. 47, No. 7, 1548--1557 (2021; Zbl 1488.93027)] for a recent survey).
Reviewer: Zdzisław Dzedzej (Gdańsk)Permanent rotations in nonholonomic mechanics. Omnirotational ellipsoidhttps://zbmath.org/1517.370652023-09-22T14:21:46.120933Z"Bizyaev, Ivan A."https://zbmath.org/authors/?q=ai:bizyaev.ivan-a"Mamaev, Ivan S."https://zbmath.org/authors/?q=ai:mamaev.ivan-sSummary: This paper is concerned with the study of permanent rotations of a rigid body rolling without slipping on a horizontal plane (i.e., the velocity of the point of contact of the ellipsoid with the plane is zero). By permanent rotations we will mean motions of a rigid body on a horizontal plane such that the angular velocity of the body remains constant and the point of contact does not change its position. A more detailed analysis is made of permanent rotations of an omnirotational ellipsoid whose characteristic feature is the possibility of permanent rotations about any point of its surface.Gyroscopic Chaplygin systems and integrable magnetic flows on sphereshttps://zbmath.org/1517.370662023-09-22T14:21:46.120933Z"Dragović, Vladimir"https://zbmath.org/authors/?q=ai:dragovic.vladimir"Gajić, Borislav"https://zbmath.org/authors/?q=ai:gajic.borislav"Jovanović, Božidar"https://zbmath.org/authors/?q=ai:jovanovic.bozidar-zarkoSummary: We introduce and study the Chaplygin systems with gyroscopic forces. This natural class of nonholonomic systems has not been treated before. We put a special emphasis on the important subclass of such systems with magnetic forces. The existence of an invariant measure and the problem of Hamiltonization are studied, both within the Lagrangian and the almost-Hamiltonian framework. In addition, we introduce problems of rolling of a ball with the gyroscope without slipping and twisting over a plane and over a sphere in \(\mathbb{R}^n\) as examples of gyroscopic \(SO (n)\)-Chaplygin systems. We describe an invariant measure and provide examples of \(SO(n-2)\)-symmetric systems (ball with gyroscope) that allow the Chaplygin Hamiltonization. In the case of additional \(SO (2)\)-symmetry, we prove that the obtained magnetic geodesic flows on the sphere \(S^{n-1}\) are integrable. In particular, we introduce the generalized Demchenko case in \(\mathbb{R}^n\), where the inertia operator of the system is proportional to the identity operator. The reduced systems are automatically Hamiltonian and represent the magnetic geodesic flows on the spheres \(S^{n-1}\) endowed with the round-sphere metric, under the influence of a homogeneous magnetic field. The magnetic geodesic flow problem on the two-dimensional sphere is well known, but for \(n>3\) was not studied before. We perform explicit integrations in elliptic functions of the systems for \(n=3\) and \(n=4\) and provide the case study of the solutions in both situations.Applications of Nijenhuis geometry. IV: Multicomponent KdV and Camassa-Holm equationshttps://zbmath.org/1517.370672023-09-22T14:21:46.120933Z"Bolsinov, Alexey V."https://zbmath.org/authors/?q=ai:bolsinov.alexey-v"Konyaev, Andrey Yu."https://zbmath.org/authors/?q=ai:konyaev.andrei-yu"Matveev, Vladimir S."https://zbmath.org/authors/?q=ai:matveev.vladimir-sSummary: We construct a new series of multi-component integrable PDE systems that contains as particular examples (with appropriately chosen parameters) and generalises many famous integrable systems including KdV, coupled KdV [\textit{M. Antonowicz} and \textit{A. P. Fordy}, Physica D 28, 345--357 (1987; Zbl 0638.35079)], Harry Dym, coupled Harry Dym [\textit{M. Antonowicz} and \textit{A. P. Fordy}, J. Phys. A, Math. Gen. 21, No. 5, L269--L275 (1988; Zbl 0673.35088)], Camassa-Holm, multicomponent Camassa-Holm [\textit{D. D. Holm} and \textit{R. I. Ivanov}, J. Phys. A, Math. Theor. 43, No. 49, Article ID 492001, 20 p. (2010; Zbl 1213.37097)], Dullin-Gottwald-Holm and Kaup-Boussinesq systems. The series also contains integrable systems with no low-component analogues.Explicit solutions and Darboux transformations of a generalized D-Kaup-Newell hierarchyhttps://zbmath.org/1517.370682023-09-22T14:21:46.120933Z"McAnally, Morgan"https://zbmath.org/authors/?q=ai:mcanally.morgan"Ma, Wen-Xiu"https://zbmath.org/authors/?q=ai:ma.wen-xiu(no abstract)A new approach to investigate the nonlinear dynamics in a \((3 + 1)\)-dimensional nonlinear evolution equation via Wronskian condition with a free functionhttps://zbmath.org/1517.370692023-09-22T14:21:46.120933Z"Wu, Jianping"https://zbmath.org/authors/?q=ai:wu.jianping(no abstract)A novel Riemann-Hilbert approach via \(t\)-part spectral analysis for a physically significant nonlocal integrable nonlinear Schrödinger equationhttps://zbmath.org/1517.370702023-09-22T14:21:46.120933Z"Wu, Jianping"https://zbmath.org/authors/?q=ai:wu.jianpingThe paper is devoted to an integrable system obtained from Manakov's system of equations, which corresponds to a two-component vector nonlinear Schrödinger equation. This is done by a reduction procedure, which essentially corresponds to imposing a nonlocal constraint that connects the two components in a consistent manner, and which results in a single equation. As a consequence, the imposed constraint should be taken into account and dealt with carefully in the subsequent analysis, which is the main difficulty to analyze the integrable properties of the system. The main interest to investigate this system, unlike the usual known integrable systems, lies in a wider range of applicability in various physical phenomena.
To investigate the integrable properties of the system the author uses the Riemann-Hilbert method, which unlike the usual cases, is considered with respect to the time variable. Surprisingly, in this case, the resulting equations turn out to be simpler in comparison to the standard case. The analytical properties of the matrices involved in the formulation of the Riemann-Hilbert problem are carefully constructed. Under the assumption that there are only simple zeros with respect to the spectral parameter, the solution to the Riemann-Hilbert problem is given, and some specific cases are constructed and discussed. In addition, the author provides some numerical analysis which confirms the analytical results.
The paper is written in a very clear and easy to follow manner, and the main proofs are given.
Reviewer: Arsen Melikyan (Brasília)The generalized Giambelli formula and polynomial KP and CKP tau-functionshttps://zbmath.org/1517.370712023-09-22T14:21:46.120933Z"Kac, Victor"https://zbmath.org/authors/?q=ai:kac.victor-g"van de Leur, Johan"https://zbmath.org/authors/?q=ai:van-de-leur.johan-wSummary: The first part of the paper is devoted to two descriptions of all polynomial tau-functions of the Kadomtsev-Petviashvili (KP) hierarchy: by a generalized Jacobi-Trudi formula, and a generalized Giambelli formula. We use the latter formula in the second part to obtain all polynomial tau-functions of the CKP hierarchy and its \(n\)-reductions. In particular, for \(n = 3\) we find all polynomial tau-functions of the Kaup-Kupershmidt hierarchy.Geometry of inhomogeneous Poisson brackets, multicomponent Harry Dym hierarchies, and multicomponent Hunter-Saxton equationshttps://zbmath.org/1517.370722023-09-22T14:21:46.120933Z"Konyaev, A. Yu."https://zbmath.org/authors/?q=ai:konyaev.andrei-yuSummary: We introduce a natural class of multicomponent local Poisson structures \(\mathcal{P}_k + \mathcal{P}_1\), where \(\mathcal{P}_1\) is a local Poisson bracket of order one and \(\mathcal{P}_k\) is a homogeneous Poisson bracket of odd order \(k\) under the assumption that \(\mathcal{P}_k\) has Darboux coordinates (Darboux-Poisson bracket) and is nondegenerate. For such brackets, we obtain general formulas in arbitrary coordinates, find normal forms (related to Frobenius triples), and provide the description of the Casimirs, using a purely algebraic procedure. In the two-component case, we completely classify such brackets up to a point transformation. From the description of Casimirs, we derive new Harry Dym (HD) hierarchies and new Hunter-Saxton (HS) equations for arbitrary number of components. In the two-component case, our HS equation differs from the well-known HS2 equation.Darboux transformations for the \(\hat{A}_{2n}^{(2)}\)-KdV hierarchyhttps://zbmath.org/1517.370732023-09-22T14:21:46.120933Z"Terng, Chuu-Lian"https://zbmath.org/authors/?q=ai:terng.chuu-lian"Wu, Zhiwei"https://zbmath.org/authors/?q=ai:wu.zhiweiAuthors' abstract: The \(\hat{A}^{(2)}_{2n}\)-hierarchy can be constructed from a splitting of the Kac-Moody algebra of type \(\hat{A}^{(1)}_{2n}\) by an involution. By choosing certain cross section of the gauge action, we obtain the \(\hat{A}^{(2)}_{2n}\)-KdV hierarchy. They are the equations for geometric invariants of isotropic curve flows of type A, which gives a geometric interpretation of the soliton hierarchy. In this paper, we construct Darboux and Bäcklund transformations for the \(\hat{A}^{(2)}_{2n}\)-hierarchy, and use it to construct Darboux transformations for the \(\hat{A}^{(2)}_{2n}\)-KdV hierarchy and isotropic curve flows of type A. Moreover, explicit soliton solutions for these hierarchies are given.
Reviewer: Ti-Jun Xiao (Fudan)Bilinear Bäcklund transformation, Lax pair and interactions of nonlinear waves for a generalized \((2 + 1)\)-dimensional nonlinear wave equation in nonlinear optics/fluid mechanics/plasma physicshttps://zbmath.org/1517.370742023-09-22T14:21:46.120933Z"Zhao, Xin"https://zbmath.org/authors/?q=ai:zhao.xin.1"Tian, Bo"https://zbmath.org/authors/?q=ai:tian.bo"Tian, He-Yuan"https://zbmath.org/authors/?q=ai:tian.he-yuan"Yang, Dan-Yu"https://zbmath.org/authors/?q=ai:yang.danyu(no abstract)Construction of abundant solutions for two kinds of \((3+1)\)-dimensional equations with time-dependent coefficientshttps://zbmath.org/1517.370752023-09-22T14:21:46.120933Z"Han, Peng-Fei"https://zbmath.org/authors/?q=ai:han.pengfei"Bao, Taogetusang"https://zbmath.org/authors/?q=ai:bao.taogetusang(no abstract)KAM tori for the two-dimensional completely resonant Schrödinger equation with the general nonlinearityhttps://zbmath.org/1517.370762023-09-22T14:21:46.120933Z"Zhang, Min"https://zbmath.org/authors/?q=ai:zhang.min.6"Si, Jianguo"https://zbmath.org/authors/?q=ai:si.jianguoThe authors deal with the quasi-periodic solutions \(u\) of the two-dimensional completely resonant Schrödinger equation with the general nonlinear term \(|u|^{2p}u\) \((p\in\mathbb{Z}^+)\) under periodic boundary conditions. They show that, for an appropriate choice of tangential sites, the considered two-dimensional Schrödinger equation has small amplitude analytic quasi-periodic solutions of specific form. To prove their result, the authors first rewrite the Schrödinger equation as a Hamiltonian system (in infinitely many coordinates) and then (using a symplectic change of coordinates) convert its Hamiltonian to a partial Birkhoff normal form, which is suitable to be treated with the help of an infinite-dimensional KAM (Kolmogorov-Arnold-Moser) theorem. The needed KAM theorem is stated and proved by a KAM iterative scheme.
Reviewer: Catalin Popa (Iaşi)KP-II approximation for a scalar Fermi-Pasta-Ulam system on a 2D square latticehttps://zbmath.org/1517.370772023-09-22T14:21:46.120933Z"Pelinovsky, Dmitry"https://zbmath.org/authors/?q=ai:pelinovsky.dmitry-e"Schneider, Guido"https://zbmath.org/authors/?q=ai:schneider.guidoSummary: We consider a scalar Fermi-Pasta-Ulam (FPU) system on a square two-dimensional lattice. The Kadomtsev-Petviashvili (KP-II) equation can be derived by means of multiple scale expansions to describe unidirectional long waves of small amplitude with slowly varying transverse modulations. We show that the KP-II approximation makes correct predictions about the dynamics of the original FPU system. An existing approximation result is extended to an arbitrary direction of wave propagation. The main novelty of this work is the use of a Fourier transform in the analysis of the FPU system in strain variables.Geometric structures on the orbits of loop diffeomorphism groups and related heavenly-type Hamiltonian systems. IIhttps://zbmath.org/1517.370782023-09-22T14:21:46.120933Z"Hentosh, O. E."https://zbmath.org/authors/?q=ai:hentosh.oksana-e"Prykarpatskyy, Ya. A."https://zbmath.org/authors/?q=ai:prykarpatsky.yarema-anatoliyovych"Balinsky, A. A."https://zbmath.org/authors/?q=ai:balinsky.alexander-a"Prykarpatski, A. K."https://zbmath.org/authors/?q=ai:prykarpatsky.anatoliy-karolevychSummary: We present a review of differential-geometric and Lie-algebraic approaches to the study of a broad class of nonlinear integrable differential systems of ``heavenly'' type associated with Hamiltonian flows on the spaces conjugated to the loop Lie algebras of vector fields on the tori. These flows are generated by the corresponding orbits of the coadjoint action of the diffeomorphism loop group and satisfy the Lax-Sato-type vector-field compatibility conditions. The corresponding hierarchies of conservation laws and their relationships with Casimir invariants are analyzed. We consider typical examples of these systems and establish their complete integrability by using the developed Lie-algebraic construction. We also describe new generalizations of the integrable dispersion-free systems of heavenly type for which the corresponding generating elements of the orbits have factorized structures, which allows their extension to the multidimensional case.
For Part I, see [the authors, Ukr. Math. J. 74, No. 8, 1175--1208 (2023; Zbl 1515.37075); translation from Ukr. Mat. Zh. 74, No. 8, 1029--1059 (2022)].Global attractors for a full von Karman beam transmission problemhttps://zbmath.org/1517.370792023-09-22T14:21:46.120933Z"Fastovska, Tamara"https://zbmath.org/authors/?q=ai:fastovska.tamaraSummary: A nonlinear transmisson problem for an elastic full von Karman beam is considered here. We prove that the system possesses a compact global attractor.Morse-Smale inequalities and Chafee-Infante attractorshttps://zbmath.org/1517.370802023-09-22T14:21:46.120933Z"Pires, Leonardo"https://zbmath.org/authors/?q=ai:pires.leonardoSummary: In this paper, we are concerned with the shape of the attractor \(\mathcal{A}^{\lambda}\) of the scalar Chafee-Infante equation. We construct a Morse-Smale vector field in the disk \(\mathbb{D}^k\) topologically equivalent to infinite-dimensional dynamics of the Chafee-Infante equation. As a consequence, we obtain geometric properties of \(\mathcal{A}^{\lambda}\) using the Morse-Smale inequalities.Exponential ergodicity for a stochastic two-layer quasi-geostrophic modelhttps://zbmath.org/1517.370812023-09-22T14:21:46.120933Z"Carigi, Giulia"https://zbmath.org/authors/?q=ai:carigi.giulia"Bröcker, Jochen"https://zbmath.org/authors/?q=ai:brocker.jochen"Kuna, Tobias"https://zbmath.org/authors/?q=ai:kuna.tobiasSummary: Ergodic properties of a stochastic medium complexity model for atmosphere and ocean dynamics are analyzed. More specifically, a two-layer quasi-geostrophic model for geophysical flows is studied, with the upper layer being perturbed by additive noise. This model is popular in the geosciences, for instance to study the effects of a stochastic wind forcing on the ocean. A rigorous mathematical analysis however meets with the challenge that in the model under study, the noise configuration is spatially degenerate as the stochastic forcing acts only on the top layer. Exponential convergence of solutions laws to the invariant measure is established, implying a spectral gap of the associated Markov semigroup on a space of Hölder continuous functions. The approach provides a general framework for generalized coupling techniques suitable for applications to dissipative SPDEs. In case of the two-layer quasi-geostrophic model, the results require the second layer to obey a certain passivity condition.Stability of pullback random attractors for stochastic 3D Navier-Stokes-Voight equations with delayshttps://zbmath.org/1517.370822023-09-22T14:21:46.120933Z"Zhang, Qiangheng"https://zbmath.org/authors/?q=ai:zhang.qianghengSummary: This paper is concerned with the limiting dynamics of stochastic retarded 3D non-autonomous Navier-Stokes-Voight (NSV) equations driven by Laplace-multiplier noise. We first prove the existence, uniqueness, forward compactness and forward longtime stability of pullback random attractors (PRAs). We then establish the upper semicontinuity of PRAs from non-autonomy to autonomy. Finally, we study the upper semicontinuity of PRAs under an analogue of Hausdorff semi-distance as the memory time tends to zero. Because of the solution has no higher regularity, the forward pullback asymptotic compactness of solutions in the state space is proved by the spectrum decomposition technique.Dispersive shock waves in lattices: a dimension reduction approachhttps://zbmath.org/1517.370832023-09-22T14:21:46.120933Z"Chong, Christopher"https://zbmath.org/authors/?q=ai:chong.christopher"Herrmann, Michael"https://zbmath.org/authors/?q=ai:herrmann.michael|herrmann.michael.1"Kevrekidis, P. G."https://zbmath.org/authors/?q=ai:kevrekidis.panayotis-gThe first goal of this paper is to provide a complementary approach to modulation theory for the description of lattice dispersive shock waves (DSWs) in a broad class of nonlinear lattice dynamical models. As further aim, the authors investigate the possibility that the lattice DSW dynamics can be reduced to a planar ODE which can be handled analytically. Two approaches are explored towards identifying the underlying ODE dynamics: a data driven one and one based on a quasi-continuum approximation. The obtained results are discussed within the context of the modulation equations
The method presented in the paper offers a key insight of relevance to a wide class of lattice models bearing DSWs and the method of analysis offers a quantitative perspective on the DSW dynamics that is found to be in good agreement with the numerical observations in suitable regimes of the wave speed.
The paper is completed with an appendix where the derivation of the modulation equations in terms of the wave parameters is presented by using Whitham's method of the averaged Lagrangian.
Reviewer: Vittorio Romano (Catania)Forecasting Hamiltonian dynamics without canonical coordinateshttps://zbmath.org/1517.370842023-09-22T14:21:46.120933Z"Choudhary, Anshul"https://zbmath.org/authors/?q=ai:choudhary.anshul"Lindner, John F."https://zbmath.org/authors/?q=ai:lindner.john-f"Holliday, Elliott G."https://zbmath.org/authors/?q=ai:holliday.elliott-g"Miller, Scott T."https://zbmath.org/authors/?q=ai:miller.scott-t"Sinha, Sudeshna"https://zbmath.org/authors/?q=ai:sinha.sudeshna"Ditto, William L."https://zbmath.org/authors/?q=ai:ditto.william-l(no abstract)Detecting causal relations in time series with the new cross Markov matrix techniquehttps://zbmath.org/1517.370852023-09-22T14:21:46.120933Z"Craciunescu, Teddy"https://zbmath.org/authors/?q=ai:craciunescu.teddy"Murari, Andrea"https://zbmath.org/authors/?q=ai:murari.andrea(no abstract)Stability analysis of chaotic systems from datahttps://zbmath.org/1517.370862023-09-22T14:21:46.120933Z"Margazoglou, Georgios"https://zbmath.org/authors/?q=ai:margazoglou.georgios"Magri, Luca"https://zbmath.org/authors/?q=ai:magri.luca(no abstract)Dynamically orthogonal Runge-Kutta schemes with perturbative retractions for the dynamical low-rank approximationhttps://zbmath.org/1517.370872023-09-22T14:21:46.120933Z"Charous, Aaron"https://zbmath.org/authors/?q=ai:charous.aaron"Lermusiaux, Pierre F. J."https://zbmath.org/authors/?q=ai:lermusiaux.pierre-f-jSummary: Whether due to the sheer size of a computational domain, the fine resolution required, or the multiples scales and stochasticity of the dynamics, the dimensionality of a system must often be reduced so that problems of interest become computationally tractable. In this paper, we develop retractions for time-integration schemes that efficiently and accurately evolve the dynamics of a system's low-rank approximation. Through differential geometry, we analyze the error incurred at each time-step due to the high-order curvature of the manifold of fixed-rank matrices. We first obtain a novel, explicit, computationally inexpensive set of algorithms that we refer to as perturbative retractions and show that the set converges to an ideal retraction that projects optimally and exactly to the manifold of fixed-rank matrices by reducing what we define as the projection-retraction error. Furthermore, each perturbative retraction itself exhibits high-order convergence to the best low-rank approximation of the full-rank solution. Using perturbative retractions, we then develop a new class of integration techniques that we refer to as dynamically orthogonal Runge-Kutta (DORK) schemes. DORK schemes integrate along the nonlinear manifold, updating the subspace upon which we project the system's dynamics as it is integrated. Through numerical test cases, we demonstrate our schemes for matrix addition, real-time data compression, and deterministic and stochastic partial differential equations. We find that DORK schemes are highly accurate by incorporating knowledge of the dynamic, nonlinear manifold's high-order curvature, and they are computationally efficient by limiting the growing rank needed to represent the evolving dynamics.Higher order extended dynamic mode decomposition based on the structured total least squareshttps://zbmath.org/1517.370882023-09-22T14:21:46.120933Z"Ding, Weiyang"https://zbmath.org/authors/?q=ai:ding.weiyang"Li, Jie"https://zbmath.org/authors/?q=ai:li.jie.16|li.jie.4|li.jie|li.jie.5|li.jie.3Summary: We develop a data-driven approach for analyzing the underlying dynamics from snapshots, which is called the higher order extended dynamic mode decomposition (HOEDMD) in this paper. The HOEDMD method, generalizing the extended dynamic mode decomposition, can handle the case when the spectral complexity of the dynamical system exceeds its spatial complexity. Moreover, the proposed method is capable of analyzing the snapshots taken from multiple trajectories by a mode-frequency-individual decomposition. We also introduce the structured total least squares technique for denoising and debiasing purposes and discuss efficient implementation details. The ability of our proposed method to accurately retrieve the modes with frequencies in linear dynamical systems is proved, which further provides an empirical choice for an optimal order. Finally, we evaluate the proposed structured total least squares based HOEDMD algorithm and apply it to four kinds of dynamical systems: a synthetic linear system to show that the proposed algorithm is less sensitive to the noises; a nonlinear dynamical system of iterates from a multilinear PageRank model to illustrate the necessity of introducing higher order cases; real-world signals for time series classification to indicate individual coefficients could parameterize trajectories and kernel tricks can be employed to enhance its performance on nonlinear real-world systems; and a real-world dynamical system of fMRI data to show the proposed algorithm retrieves modes more stably over several other dynamic mode decomposition variants.When does the method of harmonic balance give a correct prediction for mechanical systems?https://zbmath.org/1517.370892023-09-22T14:21:46.120933Z"Kogelbauer, Florian"https://zbmath.org/authors/?q=ai:kogelbauer.florian"Breunung, Thomas"https://zbmath.org/authors/?q=ai:breunung.thomasSummary: We investigate the validity of the harmonic balance method for nonlinear, multi-degree-of-freedom mechanical system with time-periodic forcing and linear damping. We provide conditions under which an approximate periodic solution obtained from this method correctly signals the existence of an actual periodic response of the full nonlinear system. These conditions improve classical results from the literature and provide a-priori computable conditions for the validity and accuracy of the harmonic balance method. Our proof is based on Newton's method in Banach spaces for an appropriately chosen functional. We also derive error bounds for the harmonic balance method and illustrate these on mechanical examples.Rich dynamics of discrete time-delayed Moran-Ricker modelhttps://zbmath.org/1517.370902023-09-22T14:21:46.120933Z"Eskandari, Z."https://zbmath.org/authors/?q=ai:eskandari.zohreh"Alidousti, J."https://zbmath.org/authors/?q=ai:alidousti.javad"Avazzadeh, Z."https://zbmath.org/authors/?q=ai:avazzadeh.zakiehSummary: The time-delayed Moran-Ricker population model is investigated in this paper with an aim to identify some of its unknown features. In this model, the decline of the essential resources arising from the previous generation emerges as a delay in the density dependency of the population. The random fluctuations in population size may cause the model's dynamics to change. In this study, we aim to scrutinize the model thoroughly and reveal more properties of the model. A discussion about the fixed points and their stability is presented in a brief way. By studying the normal form of the model through the reduction of the model to the associated center manifold, we show that the model will experience flip (period-doubling), Neimark-Sacker, strong resonances, and period-doubling-Neimark Sacker bifurcations. The bifurcation conditions are extracted with their critical coefficients. Numerical bifurcation analysis confirms the validity of theoretical findings.Qualitative analysis and optimal control strategy of an SIR model with saturated incidence and treatmenthttps://zbmath.org/1517.370912023-09-22T14:21:46.120933Z"Ghosh, Jayanta Kumar"https://zbmath.org/authors/?q=ai:ghosh.jayanta-kumar"Ghosh, Uttam"https://zbmath.org/authors/?q=ai:ghosh.uttam"Biswas, M. H. A."https://zbmath.org/authors/?q=ai:biswas.md-haider-ali"Sarkar, Susmita"https://zbmath.org/authors/?q=ai:sarkar.susmitaSummary: This paper deals with an SIR model with saturated incidence rate affected by inhibitory effect and saturated treatment function. Two control functions have been used, one for vaccinating the susceptible population and other for the treatment control of infected population. We have analysed the existence and stability of equilibrium points and investigated the transcritical and backward bifurcation. The Pontryagin's maximum principle has been used to characterize the optimal control whose numerical results show the positive impact of two controls mentioned above for controlling the disease. Efficiency analysis is also done to determine the best control strategy among vaccination and treatment.Multiple bifurcations in a discrete Bazykin predator-prey model with predator intraspecific interactions and ratio-dependent functional responsehttps://zbmath.org/1517.370922023-09-22T14:21:46.120933Z"Hu, Dongpo"https://zbmath.org/authors/?q=ai:hu.dongpo"Yu, Xiao"https://zbmath.org/authors/?q=ai:yu.xiao"Zheng, Zhaowen"https://zbmath.org/authors/?q=ai:zheng.zhaowen"Zhang, Chuan"https://zbmath.org/authors/?q=ai:zhang.chuan"Liu, Ming"https://zbmath.org/authors/?q=ai:liu.ming.5Summary: A discrete Bazykin predator-prey model with predator intraspecific interactions and ratio-dependent functional response is proposed and investigated. A brief mathematical analysis of the model involves giving fixed points and analyzing local stability. Codimension-one bifurcations such as the transcritical, fold, flip, Neimark-Sacker bifurcations and codimension-two bifurcations, including the fold-flip bifurcation, 1:2, 1:3, and 1:4 strong resonances, are investigated. Accurate theoretical analysis and exquisite numerical simulation are given simultaneously, which support the main content of this manuscript. With the help of several local attraction basins, period, and Lyapunov exponent diagrams, an intriguing set of fascinating dynamics in the two-parameter spaces of integral step size with other parameters is uncovered. The results in this paper reveal that the dynamics of the discrete-time predator-prey system in both single-parameter and two-parameter spaces are inherently rich and complex.Multifidelity robust controller design with gradient samplinghttps://zbmath.org/1517.370932023-09-22T14:21:46.120933Z"Werner, Steffen W. R."https://zbmath.org/authors/?q=ai:werner.steffen-w-r"Overton, Michael L."https://zbmath.org/authors/?q=ai:overton.michael-l"Peherstorfer, Benjamin"https://zbmath.org/authors/?q=ai:peherstorfer.benjaminSummary: Robust controllers that stabilize dynamical systems even under disturbances and noise are often formulated as solutions of nonsmooth, nonconvex optimization problems. While methods such as gradient sampling can handle the nonconvexity and nonsmoothness, the costs of evaluating the objective function may be substantial, making robust control challenging for dynamical systems with high-dimensional state spaces. In this work, we introduce multifidelity variants of gradient sampling that leverage low-cost, low-fidelity models with low-dimensional state spaces for speeding up the optimization process while nonetheless providing convergence guarantees for a high-fidelity model of the system of interest, which is primarily accessed in the last phase of the optimization process. Our first multifidelity method initiates gradient sampling on higher-fidelity models with starting points obtained from cheaper, lower-fidelity models. Our second multifidelity method relies on ensembles of gradients that are computed from low- and high-fidelity models. Numerical experiments with controlling the cooling of a steel rail profile and laminar flow in a cylinder wake demonstrate that our new multifidelity gradient sampling methods achieve up to two orders of magnitude speedup compared to the single-fidelity gradient sampling method that relies on the high-fidelity model alone.Rational maps with rational multipliershttps://zbmath.org/1517.370942023-09-22T14:21:46.120933Z"Huguin, Valentin"https://zbmath.org/authors/?q=ai:huguin.valentinSummary: In this article, we show that every rational map whose multipliers all lie in a given number field is a power map, a Chebyshev map or a Lattès map. This strengthens a conjecture by \textit{J. Milnor} [in: Dynamics on the Riemann sphere. A Bodil Branner Festschrift. Zürich: European Mathematical Society Publishing House. 9--43 (2006; Zbl 1235.37015)]
concerning rational maps with integer multipliers, which was recently proved by \textit{Z. Ji} and \textit{J. Xie} [Forum Math. Pi 11, Paper No. e11, 37 p. (2023; Zbl 07691681)].Minimally critical regular endomorphisms of \(\mathbb{A}^N\)https://zbmath.org/1517.370952023-09-22T14:21:46.120933Z"Ingram, Patrick"https://zbmath.org/authors/?q=ai:ingram.patrickSummary: We study the dynamics of the map \(f:\mathbb{A}^N\to\mathbb{A}^N\) defined by
\[
f(\mathbf{X})=A\mathbf{X}^d+\mathbf{b},
\]
for \(A\in\mathrm{SL}_N\), \(\mathbf{b}\in \mathbb{A}^N\), and \(d\geq 2\), a class which specializes to the unicritical polynomials when \(N=1\). In the case \(k=\mathbb{C}\) we obtain lower bounds on the sum of Lyapunov exponents of \(f\), and a statement which generalizes the compactness of the Mandelbrot set. Over \(\overline{\mathbb{Q}}\) we obtain estimates on the critical height of \(f\), and over algebraically closed fields we obtain some rigidity results for post-critically finite morphisms of this form.Heights and arithmetic dynamicshttps://zbmath.org/1517.370962023-09-22T14:21:46.120933Z"Yasufuku, Yu"https://zbmath.org/authors/?q=ai:yasufuku.yuSummary: We survey some recent results in the field of arithmetic dynamics. We especially focus on topics where the height functions play important roles, namely integral points in orbits and Kawaguchi-Silverman conjecture relating arithmetic degrees with dynamical degrees.Unlikely intersection problems for restricted lifts of a \(p\)-th powerhttps://zbmath.org/1517.370972023-09-22T14:21:46.120933Z"Peng, Wayne"https://zbmath.org/authors/?q=ai:peng.wayneSummary: We applied the theory of perfectoid spaces to prove dynamical versions of the Manin-Mumford conjecture, Mordell-Lang conjecture, and Tate-Voloch conjecture for lifts of a \(p\)-th power, generalizing the work of \textit{J. Xie} in [Algebra Number Theory 12, No. 7, 1715--1748 (2018; Zbl 1415.37116)] for lifts of Frobenius.Analysis of atypical orbits in one-dimensional piecewise-linear discontinuous mapshttps://zbmath.org/1517.390072023-09-22T14:21:46.120933Z"Metri, Rajanikant"https://zbmath.org/authors/?q=ai:metri.rajanikant"Rajpathak, Bhooshan"https://zbmath.org/authors/?q=ai:rajpathak.bhooshan"Pillai, Harish"https://zbmath.org/authors/?q=ai:pillai.harish-k(no abstract)The non-iterates are dense in the space of continuous self-mapshttps://zbmath.org/1517.390122023-09-22T14:21:46.120933Z"Bhat, B. V. Rajarama"https://zbmath.org/authors/?q=ai:bhat.b-v-rajarama"Gopalakrishna, Chaitanya"https://zbmath.org/authors/?q=ai:gopalakrishna.chaitanyaSummary: In this paper we first prove a nonexistence result on iterative roots, which presents several sufficient conditions for identifying self-maps on arbitrary sets that have no iterative roots of any order. Then, using this result, we prove that when \(X\) is \([0,1]^m\), \(\mathbb{R}^m\) or \(S^1\) every non-empty open set of the space \(\mathcal{C}(X)\) of continuous self-maps on \(X\) endowed with the compact-open topology contains a map that does not have even discontinuous iterative roots of order \(n\geq 2\). This, in particular, proves that the complement of \(\{f^n:f\in\mathcal{C}(X)\text{ and }n\geq 2\}\), the set of non-iterates, is dense in \(\mathcal{C}(X)\) for these \(X\).Poisson quasi-Nijenhuis deformations of the canonical PN structurehttps://zbmath.org/1517.530722023-09-22T14:21:46.120933Z"Falqui, G."https://zbmath.org/authors/?q=ai:falqui.gregorio"Mencattini, I."https://zbmath.org/authors/?q=ai:mencattini.igor"Pedroni, M."https://zbmath.org/authors/?q=ai:pedroni.marcoA bivector field \(\pi\in\Gamma(\wedge^2 TM)\) on a smooth manifold \(M\) is called a \textit{Poisson bivector} if \([\pi,\pi]_{SN}=0\), where \([\cdot,\cdot]_{SN}\) is the Schouten-Nijenhuis bracket. Such a pair \((M, \pi)\) is called a \textit{Poisson manifold}. The space of smooth fuctions \(C^\infty(M)\) on a Poisson manifold \((M, \pi)\) has a natural structure of Poisson algebra \((C^\infty(M),\{\cdot,\cdot\})\) given by
\[
\{f,g\}:=\pi(df,dg),\quad \forall f,g\in C^\infty(M).
\]
According to [\textit{Y. Kosmann-Schwarzbach} and \textit{F. Magri}, Ann. Inst. Henri Poincaré, Phys. Théor. 53, No. 1, 35--81 (1990; Zbl 0707.58048)], a \textit{Poisson Nijenhuis (PN) manifold} is given by a Poisson manifold \((M, \pi)\) together with a \((1, 1)\) tensor field \(N:TM\to TM\) such that
the Nijenhuis torsion of \(N\) vanishes, \([N,N]=0\), and
the Poisson bivector \(\pi\) is compatible with \(N\).
More generally, if a Poisson manifold \((M, \pi)\) is compatible with \(N\) and the Nijenhuis torsion of \(N\) (not necessarily vanishing) is related to a given \(3\)-form \(\phi\) on \(M\) in a specified way, such a quadruple \((M, \pi, N, \phi)\) is called a \textit{Poisson quasi-Nijenhuis (PqN) manifold}. By definition, a PqN manifold \((M, \pi, N, \phi)\) is a PN manifold iff \(\phi=0\).
The authors study deformations of PN manifolds. The main result (see Theorem 2) provides a deformation of a given PN manifold \((M, \pi, N)\) into a PqN manifold \((M, \pi, \hat{N}, \phi)\) by using a closed \(3\)-form \(\Omega\) on \(M\), where \(\phi=d_N\Omega+\frac{1}{2}[\Omega,\Omega]_\pi\).
As an application, it is shown that the canonical PN structure on \(\mathbb{R}^{2n}\) can be deformed to both the PN structure of the open (or non-periodic) \(n\)-particle Toda lattice in [\textit{A. Das} and
\textit{S. Okubo}, Ann. Phys. 190, No. 2, 215--232 (1989; \url{doi:10.1016/0003-4916(89)90014-6})], and the PqN structure of the closed (or periodic) \(n\)-particle Toda lattice, described in [\textit{G. Falqui} et al., Math. Phys. Anal. Geom. 23, No. 3, Paper No. 26, 17 p. (2020; Zbl 1453.37057)].
Reviewer: Wei Xia (Guangzhou)Semisimple flat F-manifolds in higher genushttps://zbmath.org/1517.530782023-09-22T14:21:46.120933Z"Arsie, Alessandro"https://zbmath.org/authors/?q=ai:arsie.alessandro"Buryak, Alexandr"https://zbmath.org/authors/?q=ai:buryak.alexandr-y"Lorenzoni, Paolo"https://zbmath.org/authors/?q=ai:lorenzoni.paolo"Rossi, Paolo"https://zbmath.org/authors/?q=ai:rossi.paoloFrobenius manifolds and cohomological field theories are algebraic structures motivated by Gromov-Witten theory. Frobenius manifolds capture the genus zero part of Gromov-Witten theory whereas cohmological field theories capture the all genus theory. A major tool in this area is the Givental-Teleman group action on the space of all cohomological field theories, which acts transitively on the subspace of semisimple cohomological field theories. In particular, semisimple cohomological field theories can be uniquely reconstructed from their genus zero part.
The paper generalizes these results to the more general set-up of flat F-manifolds and F-cohomological field theories. Unlike Frobenius manifolds, flat F-manifolds do not necessarily have a flat metric. Consequently, they have only a vector potential whose second derivatives give the structure constants of the product, and not a potential function whose third derivatives combined with the metric produce the structure constants in the case of Frobenius manifolds. Correspondingly, F-cohomological field theories do not necessarily satisfy the gluing axiom at nonseparating nodes. The main results of this paper are the construction of a generalization of the Givental-Teleman group action to the setting of F-cohomological field theories and the proof that this action is transitive on the subspace of semisimple F-cohomological field theories. In particular, semisimple F-cohomological field theories can be uniquely reconstructed from their genus zero part and the degree zero piece of their genus one part. The genus one information is necessary given the lack of a gluing axiom at nonseparating nodes.
Reviewer: Pierrick Bousseau (Athens)On entropy on quasi-metric spaceshttps://zbmath.org/1517.540072023-09-22T14:21:46.120933Z"Haihambo, Paulus"https://zbmath.org/authors/?q=ai:haihambo.paulus"Olela-Otafudu, Olivier"https://zbmath.org/authors/?q=ai:otafudu.olivier-olelaThe classical notion of topological entropy \(h_U\) by \textit{R. Bowen} [Trans. Am. Math. Soc. 153, 401--414 (1971; Zbl 0212.29201)] for uniformly continuous self-maps of metric spaces is extended to uniformly continuous self-maps of quasi-metric spaces. The authors call this new notion quasi-uniform entropy \(h_{QU}\) and extend the basic properties known for \(h_U\) to \(h_{QU}\). A uniformly continuous self-map \(\psi\colon(X,q)\to (X,q)\) of a quasi-metric space is also a uniformly continuous self-map of \((X,q^s)\), where \(q^s\) is the symmetrized metric of \(q\); they show that always \(h_{QU}(\psi,q)\leq h_U(\psi,q^s)\). Finally, inspired by a similar result by \textit{T. Kimura} [Commentat. Math. Univ. Carol. 39, No. 2, 389--399 (1998; Zbl 0937.54024)] for another notion of entropy, they prove what they call a completion theorem: for a join-compact quasi-metric space \((X,q)\) the quasi-uniform entropy of a self-map \(\psi\) of \((X,q)\) coincides with the quasi-uniform entropy of the extension of \(\psi\) to the bicompletion of \((X,q)\).
Reviewer: Anna Giordano Bruno (Udine)Lower bounds for eigenfunction restrictions in lacunary regionshttps://zbmath.org/1517.580092023-09-22T14:21:46.120933Z"Canzani, Yaiza"https://zbmath.org/authors/?q=ai:canzani.yaiza"Toth, John A."https://zbmath.org/authors/?q=ai:toth.john-aSummary: Let \((M, g)\) be a compact, smooth Riemannian manifold and \(\{u_h\}\) be a sequence of \(L^2\)-normalized Laplace eigenfunctions that has a localized defect measure \(\mu\) in the sense that \(M {\setminus } {{\text{supp}}}(\pi_* \mu ) \ne \emptyset\) where \(\pi :T^*M \rightarrow M\) is the canonical projection. Using Carleman estimates we prove that for any real smooth closed hypersurface \(H \subset (M{\setminus } {{\text{supp}}}\!(\pi_* \mu ))\) sufficienly close to \(\text{supp} \, \pi_* \mu\) and for all \(\delta >0\),
\[
\int_H |u_h|^2 d\sigma_{_{H}} \ge C_{\delta } e^{- [{\varphi (\tau_{_{H}})} + \delta]/h},
\]
as \(h \rightarrow 0^+\). Here, \( \varphi (\tau ) = \tau + O(\tau^2)\) and \(\tau_{_{H}}:= d(H, {{\text{supp}}}(\pi_* \mu ))\). We also show that an analogous result holds for eigenfunctions of Schrödinger operators and give applications to eigenfunctions on warped products and joint eigenfunctions of quantum completely integrable (QCI) systems.Strong approximation of Gaussian \(\beta\) ensemble characteristic polynomials: the hyperbolic regimehttps://zbmath.org/1517.600132023-09-22T14:21:46.120933Z"Lambert, Gaultier"https://zbmath.org/authors/?q=ai:lambert.gaultier"Paquette, Elliot"https://zbmath.org/authors/?q=ai:paquette.elliotIn this article, the authors study the characteristic polynomials \(\varphi_{N}\) of the Gaussian \(\beta\)-ensemble for general \(\beta>0\), and extend its known connections to log-correlated fields. In order to describe the asymptotics of the characteristic polynomial, the authors distinguish the three regimes (the hyperbolic, parabolic and elliptic) of transfer matrix recurrences of Gaussian \(\beta\)-ensemble and analyze completely the hyperbolic part of the recurrence. Applying the strong embedding theorem, the authors establish a new coupling between \(\varphi_{N}\) and a Gaussian analytic function with an error which is uniform away from the support of the semicircle law. Borrowing from the moderate deviation estimates and transfer matrix recurrence, the authors further obtain a (probabilistic) approximation for \(\varphi_{N}\) in terms of a Gaussian log-correlated field in the hyperbolic region, that is, the Gaussian \(\beta\)-ensemble characteristic polynomials can be approximated by the Hermite polynomial times the exponential of a Gaussian analytic function which arises by linearizing the transfer matrix recurrence in the hyperbolic region.
Reviewer: Guanggan Chen (Chengdu)Random walks on \(\mathrm{SL}_2({\mathbb{C}})\): spectral gap and limit theoremshttps://zbmath.org/1517.600302023-09-22T14:21:46.120933Z"Dinh, Tien-Cuong"https://zbmath.org/authors/?q=ai:tien-cuong-dinh."Kaufmann, Lucas"https://zbmath.org/authors/?q=ai:kaufmann.lucas"Wu, Hao"https://zbmath.org/authors/?q=ai:wu.hao.8Consider a probability measure \(\mu\) on the group \(\mathrm{SL}_2(\mathbb{C})\) and the corresponding random walk \((S_n)_{n\geq1}\) defined by \(S_n=g_n\cdots g_1\), where each of the \(g_i\) are independent and distributed according to \(\mu\). The authors establish local limit theorems for both the associated norm cocycle and random matrix coefficients. The first of these is established under an optimal second moment condition, and the latter requires the existence of third moments of \(\mu\). They also establish Berry-Esseen inequalities for these quantities, again under second and third moment conditions, respectively. These Berry-Esseen bounds are of order \(O(1/\sqrt{n})\). Proofs of these results rely on spectral properties of the corresponding Markov operator.
Reviewer: Fraser Daly (Edinburgh)Stochastic resonance in stochastic PDEshttps://zbmath.org/1517.600722023-09-22T14:21:46.120933Z"Berglund, Nils"https://zbmath.org/authors/?q=ai:berglund.nils"Nader, Rita"https://zbmath.org/authors/?q=ai:nader.ritaSummary: We consider stochastic partial differential equations (SPDEs) on the one-dimensional torus, driven by space-time white noise, and with a time-periodic drift term, which vanishes on two stable and one unstable equilibrium branches. Each of the stable branches approaches the unstable one once per period. We prove that there exists a critical noise intensity, depending on the forcing period and on the minimal distance between equilibrium branches, such that the probability that solutions of the SPDE make transitions between stable equilibria is exponentially small for subcritical noise intensity, while they happen with probability exponentially close to 1 for supercritical noise intensity. Concentration estimates of solutions are given in the \(H^s\) Sobolev norm for any \(s<\frac{1}{2}\). The results generalise to an infinite-dimensional setting those obtained for 1-dimensional SDEs in [\textit{N. Berglund} and \textit{B. Gentz}, Ann. Appl. Probab. 12, No. 4, 1419--1470 (2002; Zbl 1023.60052)].Statistical methods for climate scientistshttps://zbmath.org/1517.620012023-09-22T14:21:46.120933Z"DelSole, Timothy"https://zbmath.org/authors/?q=ai:delsole.timothy"Tippett, Michael K."https://zbmath.org/authors/?q=ai:tippett.michael-kPublisher's description: A comprehensive introduction to the most commonly used statistical methods relevant in atmospheric, oceanic and climate sciences. Each method is described step-by-step using plain language, and illustrated with concrete examples, with relevant statistical and scientific concepts explained as needed. Particular attention is paid to nuances and pitfalls, with sufficient detail to enable the reader to write relevant code. Topics covered include hypothesis testing, time series analysis, linear regression, data assimilation, extreme value analysis, Principal Component Analysis, Canonical Correlation Analysis, Predictable Component Analysis, and Covariance Discriminant Analysis. The specific statistical challenges that arise in climate applications are also discussed, including model selection problems associated with Canonical Correlation Analysis, Predictable Component Analysis, and Covariance Discriminant Analysis. Requiring no previous background in statistics, this is a highly accessible textbook and reference for students and early-career researchers in the climate sciences.Bayesian spatiotemporal modeling for inverse problemshttps://zbmath.org/1517.620282023-09-22T14:21:46.120933Z"Lan, Shiwei"https://zbmath.org/authors/?q=ai:lan.shiwei"Li, Shuyi"https://zbmath.org/authors/?q=ai:li.shuyi"Pasha, Mirjeta"https://zbmath.org/authors/?q=ai:pasha.mirjetaSummary: Inverse problems with spatiotemporal observations are ubiquitous in scientific studies and engineering applications. In these spatiotemporal inverse problems, observed multivariate time series are used to infer parameters of physical or biological interests. Traditional solutions for these problems often ignore the spatial or temporal correlations in the data (static model), or simply model the data summarized over time (time-averaged model). In either case, the data information that contains the spatiotemporal interactions is not fully utilized for parameter learning, which leads to insufficient modeling in these problems. In this paper, we apply Bayesian models based on spatiotemporal Gaussian processess (STGP) to inverse problems with spatiotemporal data and show that the spatial and temporal information provides more effective parameter estimation and uncertainty quantification (UQ). We demonstrate the merit of Bayesian spatiotemporal modeling for inverse problems compared with traditional static and time-averaged approaches using a time-dependent advection-diffusion partial different equation (PDE) and three chaotic ordinary differential equations (ODE). We also provide theoretic justification for the superiority of spatiotemporal modeling to fit the trajectories even if it appears cumbersome (e.g. for chaotic dynamics).The mpEDMD algorithm for data-driven computations of measure-preserving dynamical systemshttps://zbmath.org/1517.650382023-09-22T14:21:46.120933Z"Colbrook, Matthew J."https://zbmath.org/authors/?q=ai:colbrook.matthew-jSummary: Koopman operators globally linearize nonlinear dynamical systems and their spectral information is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. However, Koopman operators are infinite dimensional, and computing their spectral information is a considerable challenge. We introduce \textit{measure-preserving extended dynamic mode decomposition} (\texttt{mpEDMD}), the first Galerkin method whose eigendecomposition converges to the spectral quantities of Koopman operators for general measure-preserving dynamical systems. \texttt{mpEDMD} is a data-driven algorithm based on an orthogonal Procrustes problem that enforces measure-preserving truncations of Koopman operators using a general dictionary of observables. It is flexible and easy to use with any preexisting dynamic mode decomposition (DMD)-type method, and with different types of data. We prove convergence of \texttt{mpEDMD} for projection-valued and scalar-valued spectral measures, spectra, and Koopman mode decompositions. For the case of delay embedding (Krylov subspaces), our results include the first convergence rates of the approximation of spectral measures as the size of the dictionary increases. We demonstrate \texttt{mpEDMD} on a range of challenging examples, its increased robustness to noise compared to other DMD-type methods, and its ability to capture the energy conservation and cascade of a turbulent boundary layer flow with Reynolds number \(> 6\times 10^4\) and state-space dimension \(>10^5\).A general system of differential equations to model first-order adaptive algorithmshttps://zbmath.org/1517.650532023-09-22T14:21:46.120933Z"Belotto da Silva, Andre"https://zbmath.org/authors/?q=ai:belotto-da-silva.andre"Gazeau, Maxime"https://zbmath.org/authors/?q=ai:gazeau.maximeSummary: First-order optimization algorithms play a major role in large scale machine learning. A new class of methods, called adaptive algorithms, was recently introduced to adjust iteratively the learning rate for each coordinate. Despite great practical success in deep learning, their behavior and performance on more general loss functions are not well understood. In this paper, we derive a non-autonomous system of differential equations, which is the continuous-time limit of adaptive optimization methods. We study the convergence of its trajectories and give conditions under which the differential system, underlying all adaptive algorithms, is suitable for optimization. We discuss convergence to a critical point in the non-convex case and give conditions for the dynamics to avoid saddle points and local maxima. For convex loss function, we introduce a suitable Lyapunov functional which allows us to study its rate of convergence. Several other properties of both the continuous and discrete systems are briefly discussed. The differential system studied in the paper is general enough to encompass many other classical algorithms (such as \textsc{Heavy Ball} and \textsc{Nesterov}'s accelerated method) and allow us to recover several known results for these algorithms.Discretization of inherent ODEs and the geometric integration of DAEs with symmetrieshttps://zbmath.org/1517.650662023-09-22T14:21:46.120933Z"Kunkel, Peter"https://zbmath.org/authors/?q=ai:kunkel.peter"Mehrmann, Volker"https://zbmath.org/authors/?q=ai:mehrmann.volkerSummary: Discretization methods for differential-algebraic equations (DAEs) are considered that are based on the integration of an associated inherent ordinary differential equation (ODE). This allows to make use of any discretization scheme suitable for the numerical integration of ODEs. For DAEs with symmetries it is shown that the inherent ODE can be constructed in such a way that it inherits the symmetry properties of the given DAE and geometric properties of its flow. This in particular allows the use of geometric integration schemes with a numerical flow that has analogous geometric properties.On abelian subshiftshttps://zbmath.org/1517.683172023-09-22T14:21:46.120933Z"Karhumäki, Juhani"https://zbmath.org/authors/?q=ai:karhumaki.juhani"Puzynina, Svetlana"https://zbmath.org/authors/?q=ai:puzynina.svetlana"Whiteland, Markus A."https://zbmath.org/authors/?q=ai:whiteland.markus-aSummary: Two finite words \(u\) and \(v\) are called abelian equivalent if each letter occurs equally many times in both \(u\) and \(v\). The abelian subshift \(\mathcal A_{\boldsymbol{x}}\) of an infinite word \(\boldsymbol{x}\) is the set of infinite words \(\boldsymbol{y}\) such that, for each factor \(u\) of \(\boldsymbol{y}\), there exists a factor \(v\) of \(\boldsymbol{x}\) which is abelian equivalent to \(u\). The notion of abelian subshift gives a characterization of Sturmian words: among binary uniformly recurrent words, Sturmian words are exactly those words for which \(\mathcal A_{\boldsymbol{x}}\) equals the shift orbit closure \(\varOmega _{\boldsymbol{x}}\). On the other hand, the abelian subshift of the Thue-Morse word contains uncountably many minimal subshifts. In this paper we undertake a general study of abelian subshifts. In particular, we characterize the abelian subshifts of recurrent aperiodic balanced words and the abelian subshifts of ternary words having factor complexity \(n+2\) for all \(n\geq 1\).
For the entire collection see [Zbl 1398.68030].Parsimony as the ultimate regularizer for physics-informed machine learninghttps://zbmath.org/1517.683372023-09-22T14:21:46.120933Z"Kutz, J. Nathan"https://zbmath.org/authors/?q=ai:kutz.j-nathan"Brunton, Steven L."https://zbmath.org/authors/?q=ai:brunton.steven-lSummary: Data-driven modeling continues to be enabled by modern machine learning algorithms and deep learning architectures. The goals of such efforts revolve around the generation of models for prediction, characterization, and control of complex systems. In the context of physics and engineering, \textit{extrapolation} and \textit{generalization} are critical aspects of model discovery that are empowered by various aspects of parsimony. Parsimony can be encoded (i) in a low-dimensional coordinate system, (ii) in the representation of governing equations, or (iii) in the representation of parametric dependencies. In what follows, we illustrate techniques that leverage parsimony in deep learning to build physics-based models, culminating in a deep learning architecture that is parsimonious in coordinates and also in representing the dynamics and their parametric dependence through a simple normal form. Ultimately, we argue that promoting parsimony in machine learning results in more \textit{physical} models, i.e., models that generalize and are parametrically represented by governing equations.On some refraction billiardshttps://zbmath.org/1517.700162023-09-22T14:21:46.120933Z"De Blasi, Irene"https://zbmath.org/authors/?q=ai:de-blasi.irene"Terracini, Susanna"https://zbmath.org/authors/?q=ai:terracini.susannaSummary: The aim of this work is to continue the analysis, started in [\textit{I. De Blasi} and \textit{S. Terracini}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 218, Article ID 112766, 40 p. (2022; Zbl 1497.37109)], of the dynamics of a point-mass particle \(P\) moving in a galaxy with an harmonic biaxial core, in whose center sits a Keplerian attractive center (e.g. a Black Hole). Accordingly, the plane \(\mathbb{R}^2\) is divided into two complementary domains, depending on whether the gravitational effects of the galaxy's mass distribution or of the Black Hole prevail. Thus, solutions alternate arcs of Keplerian hyperbolæ with harmonic ellipses; at the interface, the trajectory is refracted according to Snell's law. The model was introduced in [\textit{N. Delis} et al., Mon. Not. R. Astron. Soc. 448, No. 3, 2448--2468 (2015; \url{doi:10.1093/mnras/stv064})], in view of applications to astrodynamics. In this paper we address the general issue of periodic and quasi-periodic orbits and associated caustics when the domain is a perturbation of the circle, taking advantage of KAM and Aubry-Mather theories.Correction to: ``Discrete Hamiltonian variational mechanics and Hamel's integrators''https://zbmath.org/1517.700222023-09-22T14:21:46.120933Z"Gao, Shan"https://zbmath.org/authors/?q=ai:gao.shan"Shi, Donghua"https://zbmath.org/authors/?q=ai:shi.donghua"Zenkov, Dmitry V."https://zbmath.org/authors/?q=ai:zenkov.dmitry-vCorrection to the authors' paper [ibid. 33, No. 2, Paper No. 26, 58 p. (2023; Zbl 1513.70050)].An extended continuum mixed traffic modelhttps://zbmath.org/1517.760142023-09-22T14:21:46.120933Z"Zhang, Yicai"https://zbmath.org/authors/?q=ai:zhang.yicai.1"Zhao, Min"https://zbmath.org/authors/?q=ai:zhao.min"Sun, Dihua"https://zbmath.org/authors/?q=ai:sun.dihua"Dong, Chen"https://zbmath.org/authors/?q=ai:dong.chen(no abstract)Bifurcations and exact solutions of an asymptotic rotation-Camassa-Holm equationhttps://zbmath.org/1517.760162023-09-22T14:21:46.120933Z"Liang, Jianli"https://zbmath.org/authors/?q=ai:liang.jianli"Li, Jibin"https://zbmath.org/authors/?q=ai:li.jibin"Zhang, Yi"https://zbmath.org/authors/?q=ai:zhang.yi.3(no abstract)Ray-marching Thurston geometrieshttps://zbmath.org/1517.810372023-09-22T14:21:46.120933Z"Coulon, Rémi"https://zbmath.org/authors/?q=ai:coulon.remi"Matsumoto, Elisabetta A."https://zbmath.org/authors/?q=ai:matsumoto.elisabetta-a"Segerman, Henry"https://zbmath.org/authors/?q=ai:segerman.henry"Trettel, Steve J."https://zbmath.org/authors/?q=ai:trettel.steve-jSummary: We describe algorithms that produce accurate real-time interactive in-space views of the eight Thurston geometries using ray-marching. We give a theoretical framework for our algorithms, independent of the geometry involved. In addition to scenes within a geometry \(X\), we also consider scenes within quotient manifolds and orbifolds \(X / \Gamma\). We adapt the Phong lighting model to non-Euclidean geometries. The most difficult part of this is the calculation of light intensity, which relates to the area density of geodesic spheres. We also give extensive practical details for each geometry.Heisenberg dynamics for non self-adjoint Hamiltonians: symmetries and derivationshttps://zbmath.org/1517.810442023-09-22T14:21:46.120933Z"Bagarello, F."https://zbmath.org/authors/?q=ai:bagarello.fabioSummary: In some recent literature the role of non self-adjoint Hamiltonians, \(H\neq H^\dagger\), is often considered in connection with gain-loss systems. The dynamics for these systems is, most of the times, given in terms of a Schrödinger equation. In this paper we rather focus on the Heisenberg-like picture of quantum mechanics, stressing the (few) similarities and the (many) differences with respected to the standard Heisenberg picture for systems driven by self-adjoint Hamiltonians. In particular, the role of the symmetries, \(\ast\)-derivations and integrals of motion is discussed.Superintegrability summaryhttps://zbmath.org/1517.810492023-09-22T14:21:46.120933Z"Mironov, A."https://zbmath.org/authors/?q=ai:mironov.a-v|mironov.a-p|mironov.a-b|mironov.andrei-d|mironov.artem-sergeevich|mironov.a-g|mironov.a-k|mironov.andrei-evgenevich|mironov.andrew-m|mironov.aleksei-nikolaevich|mironov.a-a|mironov.a-l"Morozov, A."https://zbmath.org/authors/?q=ai:morozov.anatolii-nikolaevich|morozov.a-v|morozov.alexander-yu|morozov.andrei-igorevich|morozov.a-g|morozov.andrew-yu|morozov.alexei-yurievich|morozov.alexandre-v|morozov.alexander-n|morozov.a-a|morozov.andrey-n|morozov.a-m|morozov.a-k|morozov.anton|morozov.albert-dmitrievich|morozov.a-c|morozov.andrei-sergeevich|morozov.andrei-alekseevichSummary: We enumerate generalizations of the superintegrability property \(<\) \textit{character} \(>\) \(\sim\) character and illuminate possible general structures behind them. We collect variations of original formulas available up to date, and emphasize the remaining difference between the cases of Hermitian and complex matrices, bosonic and fermionic ones. Especially important is that the story is in no way restricted to Gaussian potentials.Semiclassical asymptotics of oscillating tunneling for a quadratic Hamiltonian on the algebra \(\operatorname{su}(1,1)\)https://zbmath.org/1517.810562023-09-22T14:21:46.120933Z"Vybornyi, E. V."https://zbmath.org/authors/?q=ai:vybornyi.e-v"Rumyantseva, S. V."https://zbmath.org/authors/?q=ai:rumyantseva.s-vSummary: In this paper, we consider the problem of constructing semiclassical asymptotics for the tunnel splitting of the spectrum of an operator defined on an irreducible representation of the Lie algebra \(\operatorname{su}(1,1)\). It is assumed that the operator is a quadratic function of the generators of the algebra. We present coherent states and a unitary coherent transform that allow us to reduce the problem to the analysis of a second-order differential operator in the space of holomorphic functions. Semiclassical asymptotic spectral series and the corresponding wave functions are constructed as decompositions in coherent states. For some values of the system parameters, the minimal energy corresponds to a pair of nondegenerate equilibria, and the discrete spectrum of the operator has an exponentially small tunnel splitting of the levels. We apply the complex WKB method to prove asymptotic formulas for the tunnel splitting of the energies. We also show that, in contrast to the one-dimensional Schrödinger operator, the tunnel splitting in this problem not only decays exponentially but also contains an oscillating factor, which can be interpreted as tunneling interference between distinct instantons. We also show that, for some parameter values, the tunneling is completely suppressed and some of the spectral levels are doubly degenerate, which is not typical of one-dimensional systems.Random walk on quantum blobshttps://zbmath.org/1517.810702023-09-22T14:21:46.120933Z"Jadczyk, Arkadiusz"https://zbmath.org/authors/?q=ai:jadczyk.arkadiuszSummary: We describe the action of the symplectic group on the homogeneous space of squeezed states (quantum blobs) and extend this action to the semigroup. We then extend the metaplectic representation to the metaplectic (or oscillator) semigroup and study the properties of such an extension using Bargmann-Fock space. The shape geometry of squeezing is analyzed and noncommuting elements from the symplectic semigroup are proposed to be used in simultaneous monitoring of noncommuting quantum variables -- which should lead to fractal patterns on the manifold of squeezed states.Survival in two-species reaction-diffusion system with Lévy flights: renormalization group treatment and numerical simulationshttps://zbmath.org/1517.820212023-09-22T14:21:46.120933Z"Shapoval, Dmytro"https://zbmath.org/authors/?q=ai:shapoval.dmytro"Blavatska, Viktoria"https://zbmath.org/authors/?q=ai:blavatska.viktoria"Dudka, Maxym"https://zbmath.org/authors/?q=ai:dudka.maxymSummary: We analyze the two-species reaction-diffusion system including trapping reaction \(A+B\to A\) as well as coagulation/annihilation reactions \(A+A\to(A, 0)\) where particles of both species are performing Lévy flights with control parameter \(0 < \sigma < 2\), known to lead to superdiffusive behavior. The density as well as the correlation function for target particles \(B\) in such systems are known to scale with nontrivial universal exponents at space dimension \(d \leqslant d_c\). Applying the renormalization group formalism we calculate these exponents in a case of superdiffusion below the critical dimension \(d_c = \sigma\). The numerical simulations in one-dimensional case are performed as well. The quantitative estimates for the decay exponent of the density of survived particles \(B\) are in a good agreement with our analytical results. In particular, it is found that the surviving probability of the target particles in superdiffusive regime is higher than that in a system with ordinary diffusion.Global existence and lifespan of smooth solutions of biwave mapshttps://zbmath.org/1517.830042023-09-22T14:21:46.120933Z"Chiang, Yuan-Jen"https://zbmath.org/authors/?q=ai:chiang.yuanjen"Wei, Changhua"https://zbmath.org/authors/?q=ai:wei.changhuaSummary: We show that if \(f : \mathbb{R}^{1 + m} \to N\) is a biwave map from a Minkowski spacetime into a Riemannian manifold \(N\) with initial data such that the spatial dimension \(m > 6\) and the Riemannian curvature of \(N\) satisfies certain conditions, then there exists a small constant \(\varepsilon_0\) such that the biwave map \(f\) has a unique global smooth solution for \(\varepsilon \in [0, \varepsilon_0)\) (where the parameter \(\varepsilon\) indicates the size of the initial data). In case \(1 \leq m \leq 6\), we obtain the lifespan of the smooth solution of the above biwave map. Furthermore, if \(f : \mathbb{M} \to N\) is a biwave map on a Friedmann-Lemaître-Robertson-Walker spacetime under certain circumstances, we obtain the lifespan of the smooth solution of the biwave map for spatial dimension \(m \geq 1\).
{\copyright 2022 American Institute of Physics}Cross-section continuity of definitions of angular momentumhttps://zbmath.org/1517.830152023-09-22T14:21:46.120933Z"Chen, Po-Ning"https://zbmath.org/authors/?q=ai:chen.poning"Paraizo, Daniel E."https://zbmath.org/authors/?q=ai:paraizo.daniel-e"Wald, Robert M."https://zbmath.org/authors/?q=ai:wald.robert-m"Wang, Mu-Tao"https://zbmath.org/authors/?q=ai:wang.mu-tao"Wang, Ye-Kai"https://zbmath.org/authors/?q=ai:wang.ye-kai"Yau, Shing-Tung"https://zbmath.org/authors/?q=ai:yau.shing-tungSummary: We introduce a notion of `cross-section continuity' as a criterion for the viability of definitions of angular momentum, \(J\), at null infinity: If a sequence of cross-sections, \(\mathcal{C}_n\), of null infinity converges uniformly to a cross-section \(\mathcal{C}\), then the angular momentum, \(J_n\), on \(\mathcal{C}_n\) should converge to the angular momentum, \(J\), on \(\mathcal{C}\). The Dray-Streubel (DS) definition of angular momentum automatically satisfies this criterion by virtue of the existence of a well defined flux associated with this definition. However, we show that the one-parameter modification of the DS definition proposed by Compere and Nichols -- which encompasses numerous other alternative definitions -- does not satisfy cross-section continuity. On the other hand, we prove that the show that the one-parameter modification of the DS definition proposed by CoChen-Wang-Yau definition does satisfy the cross-section continuity criterion.Nonsingular black holes from conformal symmetrieshttps://zbmath.org/1517.830322023-09-22T14:21:46.120933Z"Cadoni, M."https://zbmath.org/authors/?q=ai:cadoni.mariano"Sanna, A. P."https://zbmath.org/authors/?q=ai:sanna.andrea-pSummary: We derive the form of the metric for static, nonsingular black holes with a de Sitter core, representing a deformation of the Schwarzschild solution, by assuming that the gravitational sources describe a flow between two conformal points, at small and great distances. The resulting black-hole metric turns out to be a particular case of the Fan \& Wang metric, whose parameters have been recently constrained by using the data of the S2 star orbits around the galactic center SgrA\(^\ast\).If you want to cross singularity, wrap it!https://zbmath.org/1517.830492023-09-22T14:21:46.120933Z"Nakayama, Yu"https://zbmath.org/authors/?q=ai:nakayama.yuSummary: In two-dimensional string theory, a probe D0-brane does not see the black hole singularity due to a cancellation between its metric coupling and the dilaton coupling. A similar mechanism may work in the Schwarzschild black hole in large \(D\) dimensions by considering a suitable wrapped membrane. From the asymptotic observer, the wrapped membrane looks disappearing into nothing while the continuation of the time-like trajectory beyond the singularity suggests that it would reappear as an instantaneous space-like string stretching from the singularity. A null trajectory can be extended to a null trajectory beyond the singularity. Not only the effective particle but an effective string from the wrapped membrane can exhibit the same feature.Lyapunov exponents in \(\mathcal{N} = 2\) supersymmetric Jackiw-Teitelboim gravityhttps://zbmath.org/1517.830662023-09-22T14:21:46.120933Z"Campos Delgado, Ruben"https://zbmath.org/authors/?q=ai:campos-delgado.ruben"Förste, Stefan"https://zbmath.org/authors/?q=ai:forste.stefanSummary: We study \(\mathcal{N} = 2\) supersymmetric Jackiw-Teitelboim (JT) gravity at finite temperature coupled to matter. The matter fields are related to superconformal primaries by AdS/CFT duality. Due to broken super reparametrisation invariance in the SCFT dual, there are corrections to superconformal correlators. These are generated by the exchange of super-Schwarzian modes which is dual to the exchange of 2D supergravity modes. We compute corrections to four-point functions for superconformal primaries and analyse the behaviour of out-of-time-ordered correlators. In particular, four-point functions of two pairs of primaries with mutually vanishing two-point functions are considered. By decomposing the corresponding supermultiplet into its components, we find different Lyapunov exponents. The value of the Lyapunov exponents depends on whether the correction is due to graviton, gravitini or graviphoton exchange. If mutual two-point functions do not vanish all components grow with maximal Lyapunov exponent.A temporal splitting theorem for chronological spaceshttps://zbmath.org/1517.830712023-09-22T14:21:46.120933Z"Bleybel, Ali"https://zbmath.org/authors/?q=ai:bleybel.aliSummary: We use our results concerning temporal foliations of causal sets in order to obtain a generalization of Geroch's Theorem on temporal foliations in a globally hyperbolic space-time, in the case of chronological spaces under suitable moderate requirements. Furthermore, we show that any strong causal space-time admits a ``generalized Cauchy hypersurface.''Critique of the use of geodesics in astrophysics and cosmologyhttps://zbmath.org/1517.830852023-09-22T14:21:46.120933Z"Mannheim, Philip D."https://zbmath.org/authors/?q=ai:mannheim.philip-dSummary: Since particles obey wave equations, in general one is not free to postulate that particles move on the geodesics associated with test particles. Rather, for this to be the case one has to be able to derive such behavior starting from the equations of motion that the particles obey, and to do so one can employ the eikonal approximation. To see what kind of trajectories might occur we explore the domain of support of the propagators associated with the wave equations, and extend the results of some previous propagator studies that have appeared in the literature. For a minimally coupled massless scalar field the domain of support in curved space is not restricted to the light cone, while for a conformally coupled massless scalar field the curved space domain is only restricted to the light cone if the scalar field propagates in a conformal to flat background. Consequently, eikonalization does not in general lead to null geodesics for curved space massless rays even though it does lead to straight line trajectories in flat spacetime. Equal remarks apply to the conformal invariant Maxwell equations. However, for massive particles one does obtain standard geodesic behavior this way, since they do not propagate on the light cone to begin with. Thus depending on how big the curvature actually is, in principle, even if not necessarily in practice, the standard null-geodesic-based gravitational bending formula and the general behavior of propagating light rays are in need of modification in regions with high enough curvature. We show how to appropriately modify the geodesic equations in such situations. We show that relativistic eikonalization has an intrinsic light-front structure, and show that eikonalization in a theory with local conformal symmetry leads to trajectories that are only globally conformally symmetric. Propagation of massless particles off the light cone is a curved space reflection of the fact that when light travels through a refractive medium in flat spacetime its velocity is modified from its free flat spacetime value. In the presence of gravity spacetime itself acts as a medium, and this medium can then take light rays off the light cone. This is also manifest in a conformal invariant scalar field theory propagator in two spacetime dimensions. It takes support off the light cone, doing so in fact even if the geometry is conformal to flat. We show that it is possible to obtain eikonal trajectories that are exact without approximation, and show that normals to advancing wavefronts follow these exact eikonal trajectories, with these trajectories being the trajectories along which energy and momentum are transported. In general then, in going from flat space to curved space one does not generalize flat space geodesics to curved space geodesics. Rather, one generalizes flat space wavefront normals (normals that are geodesic in flat space) to curved space wavefront normals, and in curved space normals to wavefronts do not have to be geodesic.Shilnikov chaos, low interest rates, and New Keynesian macroeconomicshttps://zbmath.org/1517.911062023-09-22T14:21:46.120933Z"Barnett, William A."https://zbmath.org/authors/?q=ai:barnett.william-a"Bella, Giovanni"https://zbmath.org/authors/?q=ai:bella.giovanni"Ghosh, Taniya"https://zbmath.org/authors/?q=ai:ghosh.taniya"Mattana, Paolo"https://zbmath.org/authors/?q=ai:mattana.paolo"Venturi, Beatrice"https://zbmath.org/authors/?q=ai:venturi.beatriceSummary: The paper shows that in a New Keynesian (NK) model, an active interest rate feedback monetary policy, when combined with a Ricardian passive fiscal policy, à la Leeper-Woodford, may induce the onset of a Shilnikov chaotic attractor in the region of the parameter space where uniqueness of the equilibrium prevails locally. Implications, ranging from long-term unpredictability to global indeterminacy, are discussed in the paper. We find that throughout the attractor, the economy lingers in particular regions, within which the emerging aperiodic dynamics tend to evolve for a long time around lower-than-targeted inflation and nominal interest rates. This can be interpreted as a liquidity trap phenomenon, produced by the existence of a chaotic attractor, and not by the influence of an unintended steady state or the Central Bank's intentional choice of a steady state nominal interest rate at its lower bound. In addition, our finding of Shilnikov chaos can provide an alternative explanation for the controversial ``loanable funds'' over-saving theory, which seeks to explain why interest rates and, to a lesser extent, inflation rates have declined to current low levels, such that the real rate of interest may be below the marginal product of capital. Paradoxically, an active interest rate feedback policy can cause nominal interest rates, inflation rates, and real interest rates unintentionally to drift downwards within a Shilnikov attractor set. Our results are robust to whether money is in the production function, in the utility function, or not in the model at all. But our results do depend upon the existence of sticky prices.Bifurcation theory of a racetrack economy in a spatial economy modelhttps://zbmath.org/1517.911532023-09-22T14:21:46.120933Z"Ikeda, Kiyohiro"https://zbmath.org/authors/?q=ai:ikeda.kiyohiro"Onda, Mikihisa"https://zbmath.org/authors/?q=ai:onda.mikihisa"Takayama, Yuki"https://zbmath.org/authors/?q=ai:takayama.yukiSummary: Racetrack economy is a conventional spatial platform for economic agglomeration in spatial economy models. Studies of this economy up to now have been conducted mostly on 2\(k\) cities, for which agglomerations proceed via so-called spatial period doubling bifurcation cascade. This paper aims at the elucidation of agglomeration mechanisms of the racetrack economy in a general setting of an arbitrary number of cities. First, an attention was paid to the existence of invariant solutions that retain their spatial distributions when the transport cost parameter is changed. A complete list of possible invariant solutions, which are inherent for replicator dynamics and are dependent on the number of cities, is presented. Next, group-theoretic bifurcation theory is used to describe bifurcation from the uniform state, thereby presenting an insightful information on spatial agglomerations. Among a plethora of theoretically possible invariant solutions, those which actually become stable for spatial economy models are obtained numerically. Asymptotic agglomeration behavior when the number of cities become very large is studied.Spread mechanism and control strategy of social network rumors under the influence of COVID-19https://zbmath.org/1517.911762023-09-22T14:21:46.120933Z"Hui, Hongwen"https://zbmath.org/authors/?q=ai:hui.hongwen"Zhou, Chengcheng"https://zbmath.org/authors/?q=ai:zhou.chengcheng"Lü, Xing"https://zbmath.org/authors/?q=ai:lu.xing"Li, Jiarong"https://zbmath.org/authors/?q=ai:li.jiarong(no abstract)Dynamics and bifurcations of a map of homographic Ricker typehttps://zbmath.org/1517.920152023-09-22T14:21:46.120933Z"Rocha, J. Leonel"https://zbmath.org/authors/?q=ai:rocha.j-leonel"Taha, Abdel-Kaddous"https://zbmath.org/authors/?q=ai:taha.abdel-kaddous|taha.abdelkaddous"Fournier-Prunaret, D."https://zbmath.org/authors/?q=ai:fournier-prunaret.daniele(no abstract)Qualitative analysis of a spatiotemporal prey-predator model with multiple Allee effect and schooling behaviourhttps://zbmath.org/1517.920162023-09-22T14:21:46.120933Z"Tiwari, Barkha"https://zbmath.org/authors/?q=ai:tiwari.barkha"Raw, S. N."https://zbmath.org/authors/?q=ai:raw.sharada-nandan|raw.sharada-nandn"Mishra, Purnedu"https://zbmath.org/authors/?q=ai:mishra.purnedu(no abstract)A systematic study of autonomous and nonautonomous predator-prey models with combined effects of fear, migration and switchinghttps://zbmath.org/1517.920172023-09-22T14:21:46.120933Z"Tiwari, Pankaj Kumar"https://zbmath.org/authors/?q=ai:tiwari.pankaj-kumar"Al Amri, Kawkab Abdullah Nabhan"https://zbmath.org/authors/?q=ai:al-amri.kawkab-abdullah-nabhan"Samanta, Sudip"https://zbmath.org/authors/?q=ai:samanta.sudip"Khan, Qamar Jalil Ahmad"https://zbmath.org/authors/?q=ai:khan.qamar-jalil-ahmad"Chattopadhyay, Joydev"https://zbmath.org/authors/?q=ai:chattopadhyay.joydev(no abstract)Intraspecific competition of predator for prey with variable rates in protected areashttps://zbmath.org/1517.920182023-09-22T14:21:46.120933Z"Tripathi, Jai Prakash"https://zbmath.org/authors/?q=ai:tripathi.jai-prakash"Jana, Debaldev"https://zbmath.org/authors/?q=ai:jana.debaldev"Vyshnavi Devi, N. S. N. V. K."https://zbmath.org/authors/?q=ai:vyshnavi-devi.n-s-n-v-k"Tiwari, Vandana"https://zbmath.org/authors/?q=ai:tiwari.vandana"Abbas, Syed"https://zbmath.org/authors/?q=ai:abbas.syed(no abstract)COVID-19 pandemic in India: a mathematical model studyhttps://zbmath.org/1517.920242023-09-22T14:21:46.120933Z"Biswas, Sudhanshu Kumar"https://zbmath.org/authors/?q=ai:biswas.sudhanshu-kumar"Ghosh, Jayanta Kumar"https://zbmath.org/authors/?q=ai:ghosh.jayanta-kumar"Sarkar, Susmita"https://zbmath.org/authors/?q=ai:sarkar.susmita"Ghosh, Uttam"https://zbmath.org/authors/?q=ai:ghosh.uttam(no abstract)Dynamical analysis of the infection status in diverse communities due to COVID-19 using a modified SIR modelhttps://zbmath.org/1517.920262023-09-22T14:21:46.120933Z"Cooper, Ian"https://zbmath.org/authors/?q=ai:cooper.ian-l|cooper.ian-a"Mondal, Argha"https://zbmath.org/authors/?q=ai:mondal.argha"Antonopoulos, Chris G."https://zbmath.org/authors/?q=ai:antonopoulos.chris-g"Mishra, Arindam"https://zbmath.org/authors/?q=ai:mishra.arindam(no abstract)A SIR forced model with interplays with the external world and periodic internal contact interplayshttps://zbmath.org/1517.920272023-09-22T14:21:46.120933Z"d'Onofrio, Alberto"https://zbmath.org/authors/?q=ai:donofrio.alberto"Duarte, Jorge"https://zbmath.org/authors/?q=ai:duarte.jorge-a"Januário, Cristina"https://zbmath.org/authors/?q=ai:januario.cristina"Martins, Nuno"https://zbmath.org/authors/?q=ai:martins.nuno-m-c|martins.nuno-f-m|martins.nuno-cAuthors' abstract: We investigated the behavior of a susceptible-infected-recovered seasonally forced model for endemic childhood infectious diseases in the case where the target population is not isolated and, moreover, fast weekly fluctuations of the social contacts occur. We considered some key scenarios of interplay of susceptible subjects with the external world, leading to subharmonic resonances and chaos. Our simulations suggest that the above-mentioned fast oscillations of the contact rate can cause the suppression/reduction of chaos and of subharmonic resonances. Thus, far from being filtered, they have an important role. If one considers an opposition of phase of the pattern of external infections w.r.t. the pattern of internal transmission rate, they result remarkably different from a scenario of synchrony. In most scenarios, the chaotic behavior is not associated to the phenomenon of the `atom-infectious', i.e. the proportion of infectious is small but not unrealistic for large populations.
Reviewer: Jiaying Zhou (Shenzhen)Zoonotic MERS-CoV transmission: modeling, backward bifurcation and optimal control analysishttps://zbmath.org/1517.920292023-09-22T14:21:46.120933Z"Ghosh, Indrajit"https://zbmath.org/authors/?q=ai:ghosh.indrajit"Nadim, Sk Shahid"https://zbmath.org/authors/?q=ai:nadim.sk-shahid"Chattopadhyay, Joydev"https://zbmath.org/authors/?q=ai:chattopadhyay.joydev(no abstract)Nonlinear dynamics of a time-delayed epidemic model with two explicit aware classes, saturated incidences, and treatmenthttps://zbmath.org/1517.920302023-09-22T14:21:46.120933Z"Goel, Kanica"https://zbmath.org/authors/?q=ai:goel.kanica"Kumar, Abhishek"https://zbmath.org/authors/?q=ai:kumar.abhishek"Nilam"https://zbmath.org/authors/?q=ai:nilam.murray-e-alexander(no abstract)Dynamics and control of COVID-19 pandemic with nonlinear incidence rateshttps://zbmath.org/1517.920392023-09-22T14:21:46.120933Z"Rohith, G."https://zbmath.org/authors/?q=ai:rohith.g"Devika, K. B."https://zbmath.org/authors/?q=ai:devika.k-b(no abstract)Autonomous and non-autonomous fixed-time leader-follower consensus for second-order multi-agent systemshttps://zbmath.org/1517.930072023-09-22T14:21:46.120933Z"Trujillo, M. A."https://zbmath.org/authors/?q=ai:trujillo.miguel-a"Aldana-López, R."https://zbmath.org/authors/?q=ai:aldana-lopez.rodrigo"Gómez-Gutiérrez, D."https://zbmath.org/authors/?q=ai:gomez-gutierrez.david"Defoort, M."https://zbmath.org/authors/?q=ai:defoort.michael"Ruiz-León, J."https://zbmath.org/authors/?q=ai:ruiz-leon.javier"Becerra, H. M."https://zbmath.org/authors/?q=ai:becerra.hector-m(no abstract)Image encryption algorithm with circle index table scrambling and partition diffusionhttps://zbmath.org/1517.940232023-09-22T14:21:46.120933Z"Zhou, Yang"https://zbmath.org/authors/?q=ai:zhou.yang.1|zhou.yang.2|zhou.yang.4|zhou.yang"Li, Chunlai"https://zbmath.org/authors/?q=ai:li.chunlai"Li, Wen"https://zbmath.org/authors/?q=ai:li.wen.5"Li, Hongmin"https://zbmath.org/authors/?q=ai:li.hongmin"Feng, Wei"https://zbmath.org/authors/?q=ai:feng.wei|feng.wei.1"Qian, Kun"https://zbmath.org/authors/?q=ai:qian.kun(no abstract)