Recent zbMATH articles in MSC 37Dhttps://zbmath.org/atom/cc/37D2024-03-13T18:33:02.981707ZWerkzeugMultiply minimal points for the product of iterateshttps://zbmath.org/1528.370032024-03-13T18:33:02.981707Z"Huang, Wen"https://zbmath.org/authors/?q=ai:huang.wen.1"Shao, Song"https://zbmath.org/authors/?q=ai:shao.song"Ye, Xiangdong"https://zbmath.org/authors/?q=ai:ye.xiangdongSummary: The multiple Birkhoff recurrence theorem states that for any \(d \in \mathbb{N}\), every system \((X, T)\) has a multiply recurrent point \(x\), i.e., \((x, x, \ldots, x)\) is recurrent under \(\tau_d =: T \times T^2 \times \cdots \times T^d\). It is natural to ask if there always is a multiply minimal point, i.e., a point \(x\) such that \((x, x, \ldots, x)\) is \(\tau_d\)-minimal. A negative answer is presented in this paper via studying the horocycle flows.
However, it is shown that for any minimal system \((X, T)\) and any nonempty open set \(U\), there is \(x \in U\) such that \(\{ n \in \mathbb{Z} :T^n x \in U, \ldots, T^{dn}x \in U\}\) is piecewise syndetic; and that for a PI minimal system, any \(M\)-subsystem of \((X^d, \tau_d)\) is minimal.Quantitative disjointness of nilflows from horospherical flowshttps://zbmath.org/1528.370052024-03-13T18:33:02.981707Z"Katz, Asaf"https://zbmath.org/authors/?q=ai:katz.asafSummary: We prove a quantitative variant of a disjointness theorem of nilflows from horospherical flows following a technique of \textit{A. Venkatesh} [Ann. Math. (2) 172, No. 2, 989--1094 (2010; Zbl 1214.11051)], combined with the structural theorems for nilflows by \textit{B. Green} et al. [Ann. Math. (2) 176, No. 2, 1231--1372 (2012; Zbl 1282.11007)].On a classification of chaotic laminations which are nontrivial basic sets of axiom a flowshttps://zbmath.org/1528.370232024-03-13T18:33:02.981707Z"Medvedev, Vladislav S."https://zbmath.org/authors/?q=ai:medvedev.vladislav-s"Zhuzhoma, E. V."https://zbmath.org/authors/?q=ai:zhuzhoma.evgenii-vSummary: We prove that, given any \(n\geqslant 3\) and \(2\leqslant q\leqslant n-1\), there is a closed \(n\)-manifold \(M^n\) admitting a chaotic lamination of codimension \(q\) whose support has the topological dimension \({n-q+1}\). For \(n=3\) and \(q=2\), such chaotic laminations can be represented as nontrivial 2-dimensional basic sets of axiom A flows on 3-manifolds. We show that there are two types of compactification (called casings) for a basin of a nonmixing 2-dimensional basic set by a finite family of isolated periodic trajectories. It is proved that an axiom A flow on every casing has repeller-attractor dynamics. For the first type of casing, the isolated periodic trajectories form a fibered link. The second type of casing is a locally trivial fiber bundle over a circle. In the latter case, we classify (up to neighborhood equivalence) such nonmixing basic sets on their casings with solvable fundamental groups. To be precise, we reduce the classification of basic sets to the classification (up to neighborhood conjugacy) of surface diffeomorphisms with one-dimensional basic sets obtained previously by \textit{V. Z. Grines} [J. Dyn. Control Syst. 6, No. 1, 97--126 (2000; Zbl 0984.37020)], \textit{R. V. Plykin} [Russ. Math. Surv. 39, No. 6, 85--131 (1984; Zbl 0584.58038); translation from Usp. Mat. Nauk 39, No. 6(240), 75--113 (1984)], \textit{A. Yu. Zhirov} [J. Dyn. Control Syst. 6, No. 3, 397--430 (2000; Zbl 1063.37035)].Equilibrium states for self-products of flows and the mixing properties of rank 1 geodesic flowshttps://zbmath.org/1528.370252024-03-13T18:33:02.981707Z"Call, Benjamin"https://zbmath.org/authors/?q=ai:call.benjamin"Thompson, Daniel J."https://zbmath.org/authors/?q=ai:thompson.daniel-jThis paper considers equilibrium states for geodesic flows on closed rank-one manifolds. It extends results found in [\textit{K.Burns} et al., Geom. Funct. Anal. 28, No. 5, 1209--1259 (2018; Zbl 1401.37038)]. The authors work on an \(n\)-dimensional closed connected \(C^\infty\) Riemannian manifold \(M\) with metric \(g\) and non-positive sectional curvature. They consider the geodesic flow \(g_t\) for real values of \(t\) on the unit tangent bundle \(T^1M\). Equilibrium states are analyzed for Hölder continuous potentials or scalar multiples of the geometric potential \(\varphi^\mu\).
The authors' main theorem is as follows: If \( g_t\) is the geodesic flow on a closed rank-1 manifold \(M\), \(\varphi\) is defined by \(\varphi : T^1M \rightarrow \mathbb{R}\), where \(\varphi = q \varphi^u\) (where \(q\) is a real number) or is Hölder continuous, and if \(P(\mathrm{Sing}, \varphi) < P(\varphi)\), then the unique equilibrium state \(\mu_\varphi\) has the Kolmogorov property. Here \(P(\phi)\) represents the topological pressure with respect to the geodesic flow and \(\mathrm{Sing}\) is the set of vectors whose rank is greater than 1. The rank of a vector \(v\) in \(T^1M\) is the dimension of the space of parallel Jacobi vector fields for the geodesic through \(v\).
The Kolmogorov property, a mixing property in ergodic theory that is somewhat weaker than Bernoulli mixing, is described in [\textit{I. P. Kornfel'd} et al., Ergodic theory. (Ehrgodicheskaya teoriya) (Russian). Moskva: ``Nauka'' (1980; Zbl 0508.28008)].
Reviewer: William J. Satzer Jr. (St. Paul)A new proof of the dimension gap for the Gauss maphttps://zbmath.org/1528.370262024-03-13T18:33:02.981707Z"Jurga, Natalia"https://zbmath.org/authors/?q=ai:jurga.nataliaSummary: In [Isr. J. Math. 124, 61--76 (2001; Zbl 1015.11040)], \textit{Y. Kifer} et al. showed that the Bernoulli measures for the Gauss map \(T(x)=1/x\bmod 1\) satisfy a `dimension gap' meaning that for some \(c > 0\), \(\sup_{\mathbf{p}}\dim\mu_{\mathbf{p}}<1-c\), where \(\mu_p\) denotes the (pushforward) Bernoulli measure for the countable probability vector \(\mathbf{p}\). In this paper we propose a new proof of the dimension gap. By using tools from thermodynamic formalism we show that the problem reduces to obtaining uniform lower bounds on the asymptotic variance of a class of potentials.Erratum: A corrected proof of the scale recurrence lemma from the paper ``Stable intersections of regular Cantor sets with large Hausdorff dimensions''https://zbmath.org/1528.370272024-03-13T18:33:02.981707Z"Moreira, Carlos"https://zbmath.org/authors/?q=ai:moreira.carlos-gustavo-t-de-a"Zamudio, Alex"https://zbmath.org/authors/?q=ai:zamudio.alex-mauricioSummary: This is an erratum for the paper by the first author and \textit{J.-C. Yoccoz} [ibid. 154, No. 1, 45--96 (2001; Zbl 1195.37015)]. We show how to fix a flaw -- a bad choice of parameters -- in the proof of the \textit{scale recurrence lemma}. This lemma is an important step towards establishing the main theorem.Upper bound on the regularity of the Lyapunov exponent for random products of matriceshttps://zbmath.org/1528.370292024-03-13T18:33:02.981707Z"Bezerra, Jamerson"https://zbmath.org/authors/?q=ai:bezerra.jamerson"Duarte, Pedro"https://zbmath.org/authors/?q=ai:duarte.pedroSummary: We prove that if \({\boldsymbol{\mu}}\) is a finitely supported measure on \({\boldsymbol{SL}}_{\mathbf{2}}({\mathbb{R}})\) with positive Lyapunov exponent but not uniformly hyperbolic, then the Lyapunov exponent function is not \({\boldsymbol{\alpha}}\)-Hölder around \({\boldsymbol{\mu}}\) for any \({\boldsymbol{\alpha}}\) exceeding the Shannon entropy of \({\boldsymbol{\mu}}\) over the Lyapunov exponent of \({\boldsymbol{\mu}}\).Bursting solutions of the Rössler equationshttps://zbmath.org/1528.370302024-03-13T18:33:02.981707Z"Fowler, A. C."https://zbmath.org/authors/?q=ai:fowler.andrew-c"McGuinness, M. J."https://zbmath.org/authors/?q=ai:mcguinness.mark-jSummary: We provide an analytic solution of the Rössler equations based on the asymptotic limit \(c\to \infty\) and we show in this limit that the solution takes the form of multiple pulses, similar to ``burst'' firing of neurons. We are able to derive an approximate Poincaré map for the solutions, which compares reasonably with a numerically derived map.The thermodynamic formalism and the central limit theorem for stochastic perturbations of circle maps with a breakhttps://zbmath.org/1528.370332024-03-13T18:33:02.981707Z"Dzhalilov, Akhtam"https://zbmath.org/authors/?q=ai:dzhalilov.akhtam-abdurakhmanovich"Mayer, Dieter"https://zbmath.org/authors/?q=ai:mayer.dieter-h"Aliyev, Abdurahmon"https://zbmath.org/authors/?q=ai:aliyev.abdurahmonSummary: Let \(T\in C^{2+\varepsilon}(S^1\setminus\{x_b\})\), \(\varepsilon>0\), be an orientation preserving circle homeomorphism with rotation number \(\rho_T=[k_1,\,k_2,\,\ldots,\,k_m,\,1,\,1,\,\ldots]\), \(m\ge1\), and a single break point \(x_b\). Stochastic perturbations \(\overline{z}_{n+1} = T(\overline{z}_n) + \sigma \xi_{n+1}\), \(\overline{z}_0:=z\in S^1\) of critical circle maps have been studied some time ago by \textit{O. Díaz-Espinosa} and \textit{R. de la Llave} [J. Mod. Dyn. 1, No. 3, 477--543 (2007; Zbl 1130.37022)], who showed for the resulting sum of random variables a central limit theorem and its rate of convergence. Their approach used the renormalization group technique. We will use here Sinai's et al. thermodynamic formalism approach, generalised to circle maps with a break point by \textit{A. Dzhalilov} et al. [Discrete Contin. Dyn. Syst. 24, No. 2, 381--403 (2009; Zbl 1168.37009)], to extend the above results to circle homemorphisms with a break point. This and the sequence of dynamical partitions allows us, following earlier work of \textit{E. B. Vul} et al. [Russ. Math. Surv. 39, No. 3, 1--40 (1984; Zbl 0561.58033); translation from Usp. Mat. Nauk 39, No. 3(237), 3--37 (1984)], to establish a symbolic dynamics for any point \({z\in S^1}\) and to define a transfer operator whose leading eigenvalue can be used to bound the Lyapunov function. To prove the central limit theorem and its convergence rate we decompose the stochastic sequence via a Taylor expansion in the variables \(\xi_i\) into the linear term \(L_n(z_0)= \xi_n+\sum\limits_{k=1}^{n-1}\xi_k\prod\limits_{j=k}^{n-1} T'(z_j)\), \(z_0\in S^1\) and a higher order term, which is possible in a neighbourhood \(A_k^n\) of the points \(z_k\), \(k\le n-1\), not containing the break points of \(T^n\). For this we construct for a certain sequence \(\{n_m\}\) a series of neighbourhoods \(A_k^{n_m}\) of the points \(z_k\) which do not contain any break point of the map \(T^{q_{n_m}}\), \(q_{n_m}\) the first return times of \(T\). The proof of our results follows from the proof of the central limit theorem for the linearized process.Anosov endomorphisms on the two-torus: regularity of foliations and rigidityhttps://zbmath.org/1528.370362024-03-13T18:33:02.981707Z"Cantarino, Marisa"https://zbmath.org/authors/?q=ai:cantarino.marisa"Varão, Régis"https://zbmath.org/authors/?q=ai:varao.regisSummary: We provide sufficient conditions for smooth conjugacy between two Anosov endomorphisms on the two-torus. From that, we also explore how the regularity of the stable and unstable foliations implies smooth conjugacy inside a class of endomorphisms including, for instance, the ones with constant Jacobian. As a consequence, we have in this class a characterisation of smooth conjugacy between special Anosov endomorphisms (defined as those having only one unstable direction for each point) and their linearisations.
{{\copyright} 2023 IOP Publishing Ltd \& London Mathematical Society}On some properties of semi-Hamiltonian systems arising in the problem of integrable geodesic flows on the two-dimensional torushttps://zbmath.org/1528.370482024-03-13T18:33:02.981707Z"Agapov, S. V."https://zbmath.org/authors/?q=ai:agapov.sergei-vadimovich"Fakhriddinov, Zh. Sh."https://zbmath.org/authors/?q=ai:fakhriddinov.zh-shSummary: \textit{M. Bialy} and \textit{A. E. Mironov} [Discrete Contin. Dyn. Syst. 29, No. 1, 81--90 (2011; Zbl 1232.37035); Nonlinearity 24, No. 12, 3541--3554 (2011; Zbl 1232.35092); J. Geom. Phys. 87, 39--47 (2015; Zbl 1304.53084)]
demonstrated in a recent series of works that the search for polynomial first integrals of a geodesic flow on the 2-torus reduces to the search for solutions to a system of quasilinear equations which is semi-Hamiltonian. We study the various properties of this system.