Recent zbMATH articles in MSC 37Chttps://zbmath.org/atom/cc/37C2024-02-15T19:53:11.284213ZWerkzeugTopological dynamics of groupoid actionshttps://zbmath.org/1526.220052024-02-15T19:53:11.284213Z"Flores, Felipe"https://zbmath.org/authors/?q=ai:flores.felipe"Măntoiu, Marius"https://zbmath.org/authors/?q=ai:mantoiu.mariusSummary: Some basic notions and results in topological dynamics are extended to continuous groupoid actions in topological spaces. We focus mainly on recurrence properties. Besides results that are analogous to the classical case of group actions, but which have to be put in the right setting, there are also new phenomena. Mostly for groupoids whose source map is not open (and there are many), some properties which were equivalent for group actions become distinct in this general framework; we illustrate this with various counterexamples.Regularity of Kleinian limit sets and Patterson-Sullivan measureshttps://zbmath.org/1526.300552024-02-15T19:53:11.284213Z"Fraser, Jonathan M."https://zbmath.org/authors/?q=ai:fraser.jonathan-mSummary: We consider several (related) notions of geometric regularity in the context of limit sets of geometrically finite Kleinian groups and associated Patterson-Sullivan measures. We begin by computing the upper and lower regularity dimensions of the Patterson-Sullivan measure, which involves controlling the relative measure of concentric balls. We then compute the Assouad and lower dimensions of the limit set, which involves controlling local doubling properties. Unlike the Hausdorff, packing, and box-counting dimensions, we show that the Assouad and lower dimensions are not necessarily given by the Poincaré exponent.On essential values of Sergeev's frequencies and exponents of oscillation for solutions of a third-order linear differential periodic equationhttps://zbmath.org/1526.340112024-02-15T19:53:11.284213Z"Stash, Aĭdamir Khazretovich"https://zbmath.org/authors/?q=ai:stash.aidamir-khazretovichSummary: In this paper, we study various types of Sergeev's frequencies and exponents of oscillation for solutions of linear homogeneous differential equations with continuous bounded coefficients. For any preassigned natural number \(N\), a periodic third-order linear differential equation is constructively built in this paper, which has the property that its upper and lower Sergeev frequency spectra of strict signs, zeros and roots, as well as the spectra of all upper and lower strong and weak oscillation indices of strict and non-strict signs, zeros, roots and hyperroots contain the same set, consisting of \(N\) different essential values, both metrically and topologically. Moreover, all these values are implemented on the same set of solutions of the constructed equation, that is, for each solution from this set, all the frequencies listed above and the oscillation exponents coincide with each other. When constructing the indicated equation and proving the required results, analytical methods of the qualitative theory of differential equations were used, in particular, methods of the theory of perturbations of solutions of linear differential equations, as well as the author's technique for controlling the fundamental system of solutions of such equations in one particular case.On the positive periodic solutions of a class of Liénard equations with repulsive singularities in degenerate casehttps://zbmath.org/1526.340212024-02-15T19:53:11.284213Z"Yu, Xingchen"https://zbmath.org/authors/?q=ai:yu.xingchen"Song, Yongli"https://zbmath.org/authors/?q=ai:song.yongli"Lu, Shiping"https://zbmath.org/authors/?q=ai:lu.shiping"Godoy, José"https://zbmath.org/authors/?q=ai:godoy.joseIn this paper, the authors analyze the existence, multiplicity and dynamics of \(T\)-periodic solutions of the Liénard equation
\[
x''+f(x)x'+h(t,x)=s,
\]
where \(f:(0,+\infty)\rightarrow\mathbb{R}\) is continuous, \(h:\mathbb{R}\times(0,+\infty)\rightarrow\mathbb{R}\) is continuous, \(T\)-periodic in the first variable and has a repulsive singularity at the origin, and \(s\) is a real parameter. Moreover, some boundedness and asymptotic assumptions on \(h\) and on the primitive function of \(f\) are made.
The main result of the paper is a classical Ambrosetti-Prodi type result for the equation above. Namely, it is proven the existence of a \(s^\ast\in\mathbb{R}\) such that the equation has \(T\)-periodic positive solutions if, and only if, \(s\leq s^\ast\). More precisely, it has, at least, one if \(s=s^\ast\) and, at least, two if \(s<s^\ast\). Furthermore, when \(s\to-\infty\), there is a \(T\)-periodic positive solution whose minimum tends to \(+\infty\) and another one whose minimum tends to 0.
Finally, some particulars cases and an application to indefinite problems are studied.
Reviewer: Eduardo Muñoz-Hernández (Madrid)Periodic solutions of discontinuous second order differential equations. The porpoising effecthttps://zbmath.org/1526.340272024-02-15T19:53:11.284213Z"Fonda, Alessandro"https://zbmath.org/authors/?q=ai:fonda.alessandro"Torres, Pedro J."https://zbmath.org/authors/?q=ai:torres.pedro-joseThe authors construct a mathematical model of the form
\[
x''+g(x)=\epsilon p(t)\tag{\(\ast\)}
\]
in order to study the so-called porpoising effect (a bouncing effect) in racing cars. Here, the perturbation \(p:{\mathbb R}\to{\mathbb R}\) is assumed to be continuous and periodic, while \(g:(a,\infty)\to{\mathbb R}\) is continuous except for a discontinuity at some \(\alpha>a\), where both the left limit \(\ell^-\) and the right limit \(\ell^+\) of \(g\) exist as finite numbers \(\ell^-<0<\ell^+\).
Under appropriate (further) assumptions on \(g\) it is shown that for sufficiently small \(|\epsilon|\), large-amplitude subharmonic solutions may arise. The proof is based on e.g.\ the Poincaré-Birkhoff theorem. In a conclusion, the authors hint to actual evidence of this phenomenon in recent Formula 1 races.
Reviewer: Christian Pötzsche (Klagenfurt)On the solvability of a periodic problem for a system of ordinary differential equations with the main positive homogeneous nonlinearityhttps://zbmath.org/1526.340282024-02-15T19:53:11.284213Z"Mukhamadiev, E."https://zbmath.org/authors/?q=ai:mukhamadiev.ergashboi-mirzoevich"Naimov, A. N."https://zbmath.org/authors/?q=ai:naimov.alizhon-nabidzhanovichLet \(P \colon \mathbb{R}^{n} \to \mathbb{R}^{n}\) be a continuous and positively homogeneous (of order \(m > 1\)) mapping. Assuming that the system \(x'(t) = P(x(t))\), \(t\in\mathbb{R}\), has no nonzero bounded solutions, the authors prove that the fact that the degree of \(P\) on the unit sphere is different from zero is a necessary and sufficient condition for the existence of an \(\omega\)-periodic solution to the periodically perturbed system
\[
x'(t) = P(x(t)) + f(t,x(t)), \quad t\in\mathbb{R},
\]
for every perturbation \(f \colon \mathbb{R}\times\mathbb{R}^{n} \to \mathbb{R}^{n}\) continuous, \(\omega\)-periodic in the first variable, and such that
\[
\lim_{|y|\to+\infty} |y|^{-m} \max_{t\in[0,\omega]} |f(t,y)|=0.
\]
This paper improves previous contributions.
Reviewer: Guglielmo Feltrin (Udine)Threshold dynamics of an almost periodic vector-borne disease modelhttps://zbmath.org/1526.340302024-02-15T19:53:11.284213Z"Zhang, Tailei"https://zbmath.org/authors/?q=ai:zhang.tailei"Zhao, Xiao-Qiang"https://zbmath.org/authors/?q=ai:zhao.xiao-qiang|zhao.xiaoqiangSummary: Many infectious diseases cannot be transmitted from human to human directly, and the transmission needs to be done via a vector. It is well known that vectors' life cycles are highly dependent on their living environment. In order to investigate dynamics of vector-borne diseases under environment influence, we propose a vector-borne disease model with almost periodic coefficients. We derive the basic reproductive number \({{\mathbf{R}}}_0\) for this model and establish a threshold type result on its global dynamics in terms of \({{\mathbf{R}}}_0\). As an illustrative example, we consider an almost periodic model of malaria transmission. Our numerical simulation results show that the basic reproductive number may be underestimated if almost periodic coefficients are replaced by their average values. Finally, we use our model to study the dengue fever transmission in Guangdong, China. The parameters are chosen to fit the reported data available for Guangdong. Numerical simulations indicate that the annual dengue fever case in Guangdong will increase steadily in the near future unless more effective control measures are implemented. Sensitivity analysis implies that the parameters with strong impact on the outcome are recovery rate, mosquito recruitment rate, mosquito mortality rate, baseline transmission rates between mosquito and human. This suggests that the effective control strategies may include intensive treatment, mosquito control, decreasing human contact number with mosquitoes (e.g., using bed nets and preventing mosquito bites), and environmental modification.Periodicity of solutions for non-autonomous neutral functional differential equations with state-dependent delayhttps://zbmath.org/1526.340442024-02-15T19:53:11.284213Z"Zhu, Jianbo"https://zbmath.org/authors/?q=ai:zhu.jianbo"Fu, Xianlong"https://zbmath.org/authors/?q=ai:fu.xianlongThe aim of this work is to prove the existence of a periodic solution for some non-autonomous neutral partial functional differential equations with state dependent delay. The phase space is axiomatically defined. The linear part is assumed to generate an evolution family. The nonlinear part is assumed to be continuous and Lipschitzian with respect to the second argument. The authors assume that the family generated by the linear part is exponentially stable. Firstly, the authors prove the existence and uniqueness of a mild solution by using Banach's fixed point theorem. Sufficient conditions are given to prove the existence of a periodic solution and its stability.
Reviewer: Khalil Ezzinbi (Marrakech)Boundary and rigidity of nonsingular Bernoulli actionshttps://zbmath.org/1526.370052024-02-15T19:53:11.284213Z"Hasegawa, Kei"https://zbmath.org/authors/?q=ai:hasegawa.kei"Isono, Yusuke"https://zbmath.org/authors/?q=ai:isono.yusuke"Kanda, Tomohiro"https://zbmath.org/authors/?q=ai:kanda.tomohiroThe paper presents a first rigidity result for Bernoulli shift actions that are not measure-preserving, indeed a significant contribution to the field. The authors achieve this by establishing solidity for specific non-singular Bernoulli actions while introducing a novel boundary concept associated with such actions.
The introduction provides an overview of the research area, emphasizing the importance of Bernoulli shift actions in understanding rigidity phenomena. Solidity, a well-studied concept in the context of probability measure-preserving (pmp) Bernoulli actions, is defined and discussed.
The authors highlight the challenge of extending results involving solidity to non-pmp Bernoulli actions, as existing proofs heavily rely on the measure-preserving condition. The main result of the states the following. Let \(G\) be a countable discrete group and consider a product measure space with two base points
\[
(\Omega , \mu):=\prod_{g\in G} (\{0,1\}, p_g\delta_0 + q_g \delta_1),
\]
where \(p_g\in (0,1)\) and \(p_g+q_g=1\) for all \(g\in G\).
Assume that \((\Omega,\mu)\) has no atoms and satisfies Kakutani's condition, so that the nonsingular Bernoulli action \(G \curvearrowright (\Omega,\mu)\) is defined. Assume further that:
(i) \(G\) is exact;
(ii) For any \(g\in G\), \(p_h = p_{gh}\) for all but finitely many \(h\in G\).
Then the Bernoulli action \(G \curvearrowright (\Omega,\mu)\) is solid.
Concrete examples are provided, showing that the result applies to groups with more than one end and groups acting on trees. Additionally, the paper introduces the concept of ``condition AO'' and discusses its role in the context of the proof, emphasizing the technical challenges associated with actions that do not preserve the measure.
Reviewer: Alcides Buss (Florianópolis)Random-like properties of chaotic forcinghttps://zbmath.org/1526.370112024-02-15T19:53:11.284213Z"Giulietti, Paolo"https://zbmath.org/authors/?q=ai:giulietti.paolo"Marmi, Stefano"https://zbmath.org/authors/?q=ai:marmi.stefano"Tanzi, Matteo"https://zbmath.org/authors/?q=ai:tanzi.matteoSummary: We prove that skew systems with a sufficiently expanding base have approximate exponential decay of correlations, meaning that the exponential rate is observed modulo an error. The fiber maps are only assumed to be Lipschitz regular and to depend on the base in a way that guarantees diffusive behaviour on the vertical component. The assumptions do not imply an hyperbolic picture and one cannot rely on the spectral properties of the transfer operators involved. The approximate nature of the result is the inevitable price one pays for having so mild assumptions on the dynamics on the vertical component. However, the error in the approximation goes to zero when the expansion of the base tends to infinity. The result can be applied beyond the original setup when combined with acceleration or conjugation arguments, as our examples show.Weighted multifractal spectrum of \(V\)-statisticshttps://zbmath.org/1526.370232024-02-15T19:53:11.284213Z"Mesón, Alejandro"https://zbmath.org/authors/?q=ai:meson.alejandro-m"Vericat, Fernando"https://zbmath.org/authors/?q=ai:vericat.fernandoSummary: We analyze and describe the weighted multifractal spectrum of \(V\)-statistics. The description will be possible when the condition of ``weighted saturation'' is fulfilled. This means that the weighted topological entropy of the set of generic points of measure \(\mu\) equals the measure-theoretic entropy of \(\mu\). \textit{C. Zhao} et al. [J. Dyn. Differ. Equations 30, No. 3, 937--955 (2018; Zbl 1422.37010)]
proved that for any ergodic measure weighted saturation is verified, generalizing a result of
\textit{R. Bowen} [Trans. Am. Math. Soc. 184, 125--136 (1974; Zbl 0274.54030)]. Here we prove that under a property of ``weighted specification'' the saturation holds for any measure. From this we obtain the description of the spectrum of \(V\)-statistics. This generalizes the variational result that \textit{A.-H. Fan} et al.
obtained for the non-weighted case [in: Further developments in fractals and related fields. Mathematical foundations and connections. Outgrowth of the 2nd international conference on fractals and related fields, Porquerolles Island, France, June 2011. New York, NY: Birkhäuser/Springer. 135--151 (2013; Zbl 1268.28014)].Poisson stable motions of monotone and strongly sub-linear non-autonomous dynamical systemshttps://zbmath.org/1526.370242024-02-15T19:53:11.284213Z"Cheban, David"https://zbmath.org/authors/?q=ai:cheban.david-nikolaiSummary: This paper is dedicated to the study of the problem of existence of Poisson stable (Bohr/Levitan almost periodic, almost automorphic, almost recurrent, recurrent, pseudo periodic, pseudo recurrent and Poisson stable) motions of monotone sub-linear non-autonomous dynamical systems. The main results we establish in the framework of general non-autonomous (cocycle) dynamical systems.
We apply our general results to the study of the problem of existence of different classes Poisson stable solutions of some types of non-autonomous evolutionary equations (Ordinary Differential Equations, Functional-Differential Equations with finite delay and Difference Equations).Topological classification of flows without heteroclinic intersections on a connected sum of manifolds \(\mathbb{S}^{n-1}\times\mathbb{S}^1 \)https://zbmath.org/1526.370252024-02-15T19:53:11.284213Z"Grines, Vyacheslav Z."https://zbmath.org/authors/?q=ai:grines.vyacheslav-z"Gurevich, Elena Ya."https://zbmath.org/authors/?q=ai:gurevich.elena-yaSummary: In this paper, we announce a result on the possibility of obtaining sufficient conditions for topological conjugacy of gradient-like flows without heteroclinic intersections, given on a connected sum of products \(S^{n-1}\times S^1\) in combinatorial terms.On classification of Morse-Smale flows on projective-like manifoldshttps://zbmath.org/1526.370262024-02-15T19:53:11.284213Z"Grines, Vyacheslav Z."https://zbmath.org/authors/?q=ai:grines.vyacheslav-z"Gurevich, Elena Ya."https://zbmath.org/authors/?q=ai:gurevich.elena-yaSummary: In this paper, the problem of topological classification of gradient-like flows without heteroclinic intersections, given on a four-dimensional projective-like manifold, is solved. We show that a complete topological invariant for such flows is a bi-color graph that describes the mutual arrangement of closures of three-dimensional invariant manifolds of saddle equilibrium states. The problem of construction of a canonical representative in each topological equivalence class is solved.A persistently singular map of \(\mathbb{T}^n\) that is \(C^1\) robustly transitivehttps://zbmath.org/1526.370272024-02-15T19:53:11.284213Z"Morelli, Juan C."https://zbmath.org/authors/?q=ai:morelli.juan-carlosThe author finds a \(C^1\) robustly transitive endomorphism displaying critical points on the \(n\)-dimensional torus. Some examples for \(C^1\) transitive maps can be found in [\textit{P. Berger} and \textit{A. Rovella}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 30, No. 3, 463--475 (2013; Zbl 1373.37096); \textit{J. Iglesias} et al., Proc. Am. Math. Soc. 144, No. 3, 1235--1250 (2016; Zbl 1354.37036); \textit{J. Iglesias} and \textit{A. Portela}, Colloq. Math. 152, No. 2, 285--297 (2018; Zbl 1393.37055); \textit{J. C. Morelli}, J. Korean Math. Soc. 58, No. 4, 977--1000 (2021; Zbl 1486.37015)].
The main result of the paper is the following:
Theorem. Given \(n \geq 2\), there exists a persistently singular endomorphism supported on \(\mathbb T^n\) that is \(C^1\) robustly transitive.
The sketch of the construction is the as follows: ``Start from an endomorphism induced by a diagonal expanding matrix with integer coefficients, with all but one direction strongly unstable and one central direction. Perturb the map to add a blending region that mixes everything getting the transitivity and then introduce artificially the critical points preserving the transitivity property.''
Reviewer: Boldizsár Kalmár (Budapest)Outer billiards outside regular polygons: tame casehttps://zbmath.org/1526.370282024-02-15T19:53:11.284213Z"Rukhovich, Philip D."https://zbmath.org/authors/?q=ai:rukhovich.philip-dSummary: We consider the periodicity problem, that is, the existence of an aperiodic point and fullness of measure of the set of periodic points for outer billiards outside regular \(n\)-gons. The lattice cases \(n=3,4,6\) are trivial: no aperiodic points exist and the set of periodic points is of full measure. The cases \(n=5,10,8,12\) (and only these cases) are regarded as tame. The periodicity problems were solved for \(n=5\) in a breakthrough paper by Tabachnikov, who pioneered a renormalization-scheme method for studying the arising self-similar structures. The case \(n=10\) is similar to \(n=5\) and was studied earlier by the present author. The present paper is devoted to the remaining cases \(n=8,12\). We establish the existence of an aperiodic orbit in outer billiards outside regular octagons and dodecagons and prove that almost all trajectories of these outer billiards are periodic. In the regular dodecagon case we give a rigorous computer-assisted proof. We establish equivalence between the outer billiards outside a regular \(n\)-gon and a regular \(n/2\)-gon, where \(n\) is even and \(n/2\) is odd. Our investigation is based on Tabachnikov's renormalization scheme.Dependence of the behaviors of trajectories of dynamic conflict systems on the interaction vectorhttps://zbmath.org/1526.370292024-02-15T19:53:11.284213Z"Satur, O. R."https://zbmath.org/authors/?q=ai:satur.o-rSummary: We study several models of dynamic conflict systems. Their behaviors are characterized by a certain quantity called an interaction vector. The interaction vector determines the dynamics of the entire system and its limit states. The existence of the equilibrium limit states of these systems is proved and the conditions for the existence of limit cycles are established. The nonlinear dynamics of the system is illustrated by specific computer examples.Smooth ergodic theory of \(\mathbb{Z}^d \)-actionshttps://zbmath.org/1526.370302024-02-15T19:53:11.284213Z"Brown, Aaron"https://zbmath.org/authors/?q=ai:brown.aaron-w"Hertz, Federico Rodriguez"https://zbmath.org/authors/?q=ai:rodriguez-hertz.federico"Wang, Zhiren"https://zbmath.org/authors/?q=ai:wang.zhirenSummary: In the first part of this paper, we formulate a general setting in which to study the smooth ergodic theory of differentiable \(\mathbb{Z}^d\)-actions preserving a Borel probability measure. This framework includes actions by \(C^{1+{\text{Hölder}}}\) diffeomorphisms of compact manifolds. We construct intermediate unstable manifolds and coarse Lyapunov manifolds for the action as well as establish controls on their local geometry. In the second part, we consider the relationship between entropy, Lyapunov exponents, and the geometry of conditional measures for rank-1 systems given by a number of generalizations of the Ledrappier-Young entropy formulas. In the third part, for a smooth action of \(\mathbb{Z}^d\) preserving a Borel probability measure, we show that entropy satisfies a certain ``product structure'' along coarse unstable manifolds. Moreover, given two smooth \(\mathbb{Z}^d\)-actions -- one of which is a measurable factor of the other -- we show that all coarse-Lyapunov exponents contributing to the entropy of the factor system are coarse Lyapunov exponents of the total system. As a consequence, we derive an Abramov-Rohlin formula for entropy subordinated to coarse Lyapunov manifolds.Limit sets and global dynamic for 2-D divergence-free vector fieldshttps://zbmath.org/1526.370312024-02-15T19:53:11.284213Z"Marzougui, Habib"https://zbmath.org/authors/?q=ai:marzougui.habibSummary: The global structure of divergence-free vector fields on closed surfaces is investigated. We prove that if \(M\) is a closed surface and \(\mathcal{V}\) is a divergence-free \(C^1\)-vector field with finitely many singularities on \(M\) then every orbit \(L\) of \(\mathcal{V}\) is one of the following types: (i) a singular point, (ii) a periodic orbit, (iii) a closed (non periodic) orbit in \(M^*=M-\text{Sing}(\mathcal{V})\), (iv) a locally dense orbit, where Sing\((\mathcal{V})\) denotes the set of singular points of \(\mathcal{V}\). On the other hand, we show that the complementary in \(M\) of periodic components and minimal components is a compact invariant subset consisting of singularities and closed (non compact) orbits in \(M^*\). These results extend those of \textit{T. Ma} and \textit{S. Wang} [Discrete Contin. Dyn. Syst. 7, No. 2, 431--445 (2001; Zbl 1013.37049)] established when the divergence-free vector field \(\mathcal{V}\) is regular that is all its singular points are non-degenerate.Perturbations with nonpositive Lyapunov exponentshttps://zbmath.org/1526.370322024-02-15T19:53:11.284213Z"Barreira, Luis"https://zbmath.org/authors/?q=ai:barreira.luis-m"Valls, Claudia"https://zbmath.org/authors/?q=ai:valls.claudiaThe authors' work focuses on a classical feature of hyperbolicity of dynamical systems, expressed in terms of admissibility. An example of the connection between hyperbolicity and admissibility is the existence of bounded solutions under bounded perturbations. As it is known, hyperbolicity plays an important role in the theory of stability of dynamical systems.
Specifically in this paper, the authors propose a new characterization of nonautonomous discrete-time dynamics. The proposed characterization relies on the property that any perturbed dynamics with a non-positive Lyapunov exponent has a solution with a non-positive Lyapunov exponent.
Reviewer: Anatoly Martynyuk (Kyïv)Quenched and annealed equilibrium states for random Ruelle expanding maps and applicationshttps://zbmath.org/1526.370342024-02-15T19:53:11.284213Z"Stadlbauer, Manuel"https://zbmath.org/authors/?q=ai:stadlbauer.manuel"Varandas, Paulo"https://zbmath.org/authors/?q=ai:varandas.paulo"Zhang, Xuan"https://zbmath.org/authors/?q=ai:zhang.xuanSummary: We find generalized conformal measures and equilibrium states for random dynamics generated by Ruelle expanding maps, under which the dynamics exhibits exponential decay of correlations. This extends results by \textit{V. Baladi} [Commun. Math. Phys. 186, No. 3, 671--700 (1997; Zbl 0884.60097)] and \textit{M. Carvalho} et al. [J. Stat. Phys. 166, No. 1, 114--136 (2017; Zbl 1379.37073)], where the randomness is driven by an independent and identically distributed process and the phase space is assumed to be compact. We give applications in the context of weighted non-autonomous iterated function systems, free semigroup actions and introduce a boundary of equilibria for not necessarily free semigroup actions.Hyperbolicity of maximal entropy measures for certain maps isotopic to Anosov diffeomorphismshttps://zbmath.org/1526.370352024-02-15T19:53:11.284213Z"Álvarez, Carlos F."https://zbmath.org/authors/?q=ai:alvarez.carlos-fConsider a continuous transformation \(f : M \to M\), where \(M\) is a compact metric space. An ergodic invariant probability measure \(\mu\) such that \(h_{\mu}(f) = h_{\mathrm{top}}(f)\) is called a \textit{maximal entropy measure} for \(f\), where \(h_{\mu}(f)\) is the entropy with respect to \(\mu\) and \(h_{\mathrm{top}}(f)\) is the topological entropy. Such measures describe the complexity of the whole system.
The author shows that for a class of partially hyperbolic diffeomorphisms on \(\mathbb{T}^d\), which have compact two-dimensional center foliations, the maximal entropy measures (if exist) are \textit{hyperbolic}, that is, they have no zero Lyapunov exponents and there exist Lyapunov exponents with different signs.
Reviewer: Deliang Chen (Wenzhou)On the number of ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms with compact center leaveshttps://zbmath.org/1526.370362024-02-15T19:53:11.284213Z"Rocha, Joas Elias"https://zbmath.org/authors/?q=ai:rocha.joas-elias"Tahzibi, Ali"https://zbmath.org/authors/?q=ai:tahzibi.aliSummary: In this paper we study the number of ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms defined on 3-torus with compact center leaves. Assuming the existence of a periodic leaf with Morse-Smale dynamics we prove a sharp upper bound for the number of maximal measures in terms of the number of sources and sinks of Morse-Smale dynamics. A well-known class of examples for which our results apply are the so called Kan-type diffeomorphisms admitting physical measures with intermingled basins.On codimension one partially hyperbolic diffeomorphismshttps://zbmath.org/1526.370372024-02-15T19:53:11.284213Z"Zhang, Xiang"https://zbmath.org/authors/?q=ai:zhang.xiang.5|zhang.xiang.3The author considers \(C^1\)-diffeomorphisms on \(C^\infty\)-closed Riemannian manifolds \(M\) of dimension \(n\). Such diffeomorphisms \(f: M \rightarrow M\) are called codimension-one partially hyperbolic diffeomorphisms if \(f\) or \(f^{-1}\) allows a continuous \(Tf\)-invariant splitting \(TM = E^c \bigoplus E^u\), where \(E^c\) is one-dimensional, and there is a function \(\xi: M \rightarrow (1, \infty)\) such that \(||Tf(v^c)|| < \xi(x) < ||Tf(v^u)||\) for all \(x \in M\) and unit vectors \(v^c \in E^c_x\) and \(v^u \in E^u_x\).
The author shows that every codimension-one partially hyperbolic diffeomorphism arises from a linear codimension-one Anosov diffeomorphism of the torus. More specifically, the main result is the following: If \(N\) is a closed Riemannian manifold, and \(f: N \rightarrow N\) is a codimension-one partially hyperbolic diffeomorphism, then \(N\) is homeomorphic to the \(n\)-torus \(\mathbb{T}^n\), the distribution \(E^c\) is locally uniquely integrable, and \(f\) is semiconjugated to a linear codimension one Anosov diffeomorphism \(f_* \in \mathrm{GL}(n, \mathbb{Z})\), the induced linear transformation of \(f\) on \(\pi_1(N)\).
This provides a generalization of results in [\textit{J. Franks}, Proc. Symp. Pure Math. 14, 61-- 93 (1970; Zbl 0207.54304); \textit{S.E. Newhouse}, Am. J. Math. 92, 761--770 (1970; Zbl 0204.56901)].
Reviewer: William J. Satzer Jr. (St. Paul)Nonlinear thermodynamic formalism for flowshttps://zbmath.org/1526.370382024-02-15T19:53:11.284213Z"Barreira, Luis"https://zbmath.org/authors/?q=ai:barreira.luis-m"Holanda, Carllos Eduardo"https://zbmath.org/authors/?q=ai:holanda.carllosSummary: We introduce a version of the nonlinear thermodynamic formalism for flows. Moreover, we discuss the existence, uniqueness, and characterization of equilibrium measures for almost additive families of continuous functions with tempered variation. We also consider with some care the special case of additive families for which it is possible to strengthen some of the results. The proofs are mainly based on multifractal analysis.Uniform distribution of saddle connection lengths in all \(\mathrm{SL}(2,\mathbb{R})\) orbitshttps://zbmath.org/1526.370412024-02-15T19:53:11.284213Z"Robertson, Donald"https://zbmath.org/authors/?q=ai:robertson.donald"Dozier, Benjamin"https://zbmath.org/authors/?q=ai:dozier.benjaminSummary: For every flat surface, almost every flat surface in its \(\mathrm{SL}(2,\mathbb{R})\) orbit has the following property: the sequence of its saddle connection lengths in non-decreasing order is uniformly distributed in the unit interval.Knot as a complete invariant of the diffeomorphism of surfaces with three periodic orbitshttps://zbmath.org/1526.370482024-02-15T19:53:11.284213Z"Baranov, D. A."https://zbmath.org/authors/?q=ai:baranov.denis-alekseevich"Kosolapov, E. S."https://zbmath.org/authors/?q=ai:kosolapov.e-s"Pochinka, O. V."https://zbmath.org/authors/?q=ai:pochinka.olga-vSummary: It is known that Morse-Smale diffeomorphisms with two hyperbolic periodic orbits exist only on the sphere and they are all topologically conjugate to each other. However, if we allow three orbits to exist then the range of manifolds admitting them widens considerably. In particular, the surfaces of arbitrary genus admit such orientation-preserving diffeomorphisms. In this article we find a complete invariant for the topological conjugacy of Morse-Smale diffeomorphisms with three periodic orbits. The invariant is completely determined by the homotopy type (a pair of coprime numbers) of the torus knot which is the space of orbits of an unstable saddle separatrix in the space of orbits of the sink basin. We use the result to calculate the exact number of the topological conjugacy classes of diffeomorphisms under consideration on a given surface as well as to relate the genus of the surface to the homotopy type of the knot.\(C^0\)-gap between entropy-zero Hamiltonians and autonomous diffeomorphisms of surfaceshttps://zbmath.org/1526.370492024-02-15T19:53:11.284213Z"Brandenbursky, Michael"https://zbmath.org/authors/?q=ai:brandenbursky.michael"Khanevsky, Michael"https://zbmath.org/authors/?q=ai:khanevsky.michaelThis paper considers a long-standing conjecture by \textit{D. V. Anosov} and \textit {A. B. Katok} [Trudy Moskovskogo Matematiceskogo Obscestva 23, 3--36 (1970; Zbl 0218.58008)] that asks whether every entropy-zero Hamiltonian diffeomorphism of a surface lies in the \(C^0\)-closure of a set of integrable diffeomorphisms. Here the authors explore a slightly weaker conjecture: Every entropy-zero Hamiltonian diffeomorphism lies in the \(C^0\)-closure of the set of autonomous diffeomorphisms.
The authors show that the latter conjecture is false. Instead they prove that on a surface \(\Sigma\) the set of autonomous Hamiltonian diffeomorphisms is not \(C^0\)-dense in the set of zero-entropy Hamiltonians. They construct an entropy-zero Hamiltonian diffeomorphism on \(\Sigma\) that is the composition of two autonomous Hamiltonian diffeomorphisms supported on an annulus, that is symplectically embedded in \(\Sigma\) and cannot be the \(C^0\)-limit of autonomous diffeomorphisms.
Reviewer: William J. Satzer Jr. (St. Paul)Flows with minimal number of singularities in the Boy's surfacehttps://zbmath.org/1526.370512024-02-15T19:53:11.284213Z"Prishlyak, Alexandr"https://zbmath.org/authors/?q=ai:prishlyak.alexandr-olegovich"Di Beo, Luca"https://zbmath.org/authors/?q=ai:di-beo.lucaSummary: We study flows on the Boy's surface. The Boy's surface is the image of the projective plane under a certain immersion into the three-dimensional Euclidean space. It has a natural stratification consisting of one 0-dimensional stratum (central point), three 1-dimensional strata (loops starting at this point), and four 2-dimensional strata (three of them are disks lying on the same plane as the 1-dimensional strata, and having the loops as boundaries). We found all 342 optimal Morse-Smale flows and all 80 optimal projective Morse-Smale flows on the Boy's surface.Actions of symplectic homeomorphisms/diffeomorphisms on foliations by curves in dimension 2https://zbmath.org/1526.370522024-02-15T19:53:11.284213Z"Arnaud, Marie-Claude"https://zbmath.org/authors/?q=ai:arnaud.marie-claude"Zavidovique, Maxime"https://zbmath.org/authors/?q=ai:zavidovique.maximeThe authors study foliations by curves in a two-dimensional symplectic setting. The two main results deal with the regularity of the invariant foliation of a \(C^0\)-integrable symplectic twist diffeomorphism of the two-dimensional annulus in the following two cases:
(1) The generating function of such a foliation is \(C^1\);
(2) The foliation is Hölder with exponent \(\frac{1}{2}\).
In other words, the authors want to answer the following questions:
(a) When is the given foliation (locally or globally) symplectically homeomorphic to the straight foliation?
(b) What can be said about the foliation when it is invariant by a symplectic twist?
(c) What can be said about the symplectic dynamics that preserves such a foliation?
The conditions for the existence of Arnold-Liouville coordinates are also investigated.
Reviewer: Manuel de León (Madrid)Multiplicities of limit cycles appearing after perturbations of hyperbolic polycycleshttps://zbmath.org/1526.370632024-02-15T19:53:11.284213Z"Dukov, Andrei V."https://zbmath.org/authors/?q=ai:dukov.andrei-vSummary: The problem of the multiplicity of limit cycles appearing after a perturbation of a hyperbolic polycycle with generic set of characteristic numbers is considered. In particular, it is proved that the multiplicity of any limit cycle appearing after a perturbation in a smooth finite-parameter family does not exceed the number of separatrix connections forming the polycycle.Quasimonotone random and stochastic functional differential equations with applicationshttps://zbmath.org/1526.370652024-02-15T19:53:11.284213Z"Bai, Xiaoming"https://zbmath.org/authors/?q=ai:bai.xiaoming"Jiang, Jifa"https://zbmath.org/authors/?q=ai:jiang.jifa"Xu, Tianyuan"https://zbmath.org/authors/?q=ai:xu.tianyuan|xu.tianyuan.1Summary: In this paper, we study monotone properties of random and stochastic functional differential equations and their global dynamics. First, we show that random functional differential equations (RFDEs) generate the random dynamical system (RDS) if and only if all the solutions are globally defined, and establish the comparison theorem for RFDEs and the random Riesz representation theorem. These three results lead to the Borel measurability of coefficient functions in the Riesz representation of variational equations for quasimonotone RFDEs, which paves the way following the Smith line to establish eventual strong monotonicity for the RDS under cooperative and irreducible conditions. Then strong comparison principles, strong sublinearity theorems and the existence of random attractors for RFDEs are proved. Finally, criteria are presented for the existence of a unique random equilibrium and its global stability in the universe of all the tempered random closed sets of the positive cone. Applications to typical random or stochastic delay models in monotone dynamical systems, such as biochemical control circuits, cyclic gene models and Hopfield-type neural networks, are given.Evolutionary force billiardshttps://zbmath.org/1526.370702024-02-15T19:53:11.284213Z"Fomenko, Anatoly T."https://zbmath.org/authors/?q=ai:fomenko.anatolii-t"Vedyushkina, Viktoriya V."https://zbmath.org/authors/?q=ai:vedyushkina.viktoriya-viktorovnaSummary: A new class of integrable billiards has been introduced: evolutionary force billiards. They depend on a parameter and change their topology as energy (time) increases. It has been proved that they realize some important integrable systems with two degrees of freedom on the entire symplectic four-dimensional phase manifold at a time, rather than on only individual isoenergy 3-surfaces. For instance, this occurs in the Euler and Lagrange cases. It has also been proved that these two well-known systems are ``billiard-equivalent'', despite the fact that the former one is square integrable, and the latter one allows a linear integral.Integrability in a nonlinear model of swing oscillatory motionhttps://zbmath.org/1526.370732024-02-15T19:53:11.284213Z"Nikolov, Svetoslav G."https://zbmath.org/authors/?q=ai:nikolov.svetoslav-g"Vassilev, Vassil M."https://zbmath.org/authors/?q=ai:vassilev.vassil-mIn this paper, the integrable cases of dynamical systems describing the rider and the swing pumped as a compound pendulum are investigated.
In the introduction, the problem is presented (see also [\textit{W. Case} and \textit{M. Swanson}, Am. J. Phys. 58, No. 5, 463--467 (1990; \url{doi:10.1119/1.16477})]). In the appendix, the expressions for the kinetic and potential energies are deduced.
In Section 2, the equations of motion given in the Lagrangian formulation are given.
In Section 3, the Hamiltonian equations are obtained.
In Section 4, the main results of the paper are given. The integrability of the Hamiltonian equations is then discussed. Two integrable cases are obtained and the corresponding dynamical features are studied.
Reviewer: Cristian Lăzureanu (Timişoara)Realization of geodesic flows with a linear first integral by billiards with slippinghttps://zbmath.org/1526.370762024-02-15T19:53:11.284213Z"Vedyushkina, Viktoriya V."https://zbmath.org/authors/?q=ai:vedyushkina.viktoriya-viktorovna"Zav'yalov, Vladimir N."https://zbmath.org/authors/?q=ai:zavyalov.vladimir-nSummary: An arbitrary geodesic flow on the projective plane or Klein bottle with an additional, linear in the momentum, first integral is modelled using billiards with slipping on table complexes. The requisite table of a circular topological billiard with slipping is constructed algorithmically. Furthermore, linear integrals of geodesic flows can be reduced to the same canonical integral of a circular planar billiard.Dynamic and thermodynamic stability of charged perfect fluid starshttps://zbmath.org/1526.830062024-02-15T19:53:11.284213Z"Shi, Kai"https://zbmath.org/authors/?q=ai:shi.kai"Tian, Yu"https://zbmath.org/authors/?q=ai:tian.yu.1"Wu, Xiaoning"https://zbmath.org/authors/?q=ai:wu.xiaoning"Zhang, Hongbao"https://zbmath.org/authors/?q=ai:zhang.hongbao"Zhang, Jingchao"https://zbmath.org/authors/?q=ai:zhang.jingchaoSummary: We perform a thorough analysis of the dynamic and thermodynamic stability for the charged perfect fluid star by applying the Wald formalism to the Lagrangian formulation of Einstein-Maxwell-charged fluid system. As a result, we find that neither the presence of the additional electromagnetic field nor the Lorentz force experienced by the charged fluid makes any obstruction to the key steps towards the previous results obtained for the neutral perfect fluid star. Therefore, the criterion for the dynamic stability of our charged star in dynamic equilibrium within the symplectic complement of the trivial perturbations with the Arnowitt-Deser-Misner (ADM) 3-momentum unchanged is given by the non-negativity of the canonical energy associated with the timelike Killing field, where it is further shown for both non-axisymmetric and axisymmetric perturbations that the dynamic stability against these restricted perturbations also implies the dynamic stability against more generic perturbations. On the other hand, the necessary condition for the thermodynamic stability of our charged star in thermodynamic equilibrium is given by the positivity of the canonical energy of all the linear on-shell perturbations with the ADM angular momentum unchanged in the comoving frame, which is equivalent to the positivity of the canonical energy associated with the timelike Killing field when restricted onto the axisymmetric perturbations. As a by-product, we further establish the equivalence of the dynamic and thermodynamic stability with respect to the spherically symmetric perturbations of the static, spherically symmetric isentropic charged star.