Consider a topologically transitive countable Markov shift $Σ$ and a summable Markov potential $ϕ$ with finite Gurevich pressure and $\mathrm{Var}_1(ϕ) < \infty$. We prove existence of the limit $\lim_{t \to \infty} μ_t$ in the weak$^\star$ topology, where $μ_t$ is the unique equilibrium state associated to the potential $tϕ$. Besides that, we present examples where the limit at zero temperature exists for potentials satisfying more general conditions.
Consider a topologically transitive countable Markov shift and, let $f$ be a summable potential with bounded variation and finite Gurevic pressure. We prove that there exists an equilibrium state $μ_{tf}$ for each $t > 1$ and that there exists accumulation points for the family $(μ_{tf})_{t>1}$ as $t \to \infty$. We also prove that the Kolmogorov-Sinai entropy is continuous at $\infty$ with respect to the parameter $t$, that is $\lim_{t \to \infty} h(μ_{tf})=h(μ_{\infty})$, where $μ_{\infty}$ is an accumulation point of the family $(μ_{tf})_{t>1}$. These results do not depend on the existence …
In this report, we first recall the Poincaré's classification theorem for minimal orientation-preserving homeomorphisms on the circle and the Ghys' classification theorem for minimal orientation-preserving group actions on the circle. Then we introduce a classification theorem for a specified class of topologically transitive orientation-preserving group actions on the circle by $\mathbb Z^d$. Also, some groups that admit/admit no topologically transitive actions on the line are determined.
It is well known that the space of invariant probability measures for transitive sub-shifts of finite type is a Poulsen simplex. In this article we prove that in the non-compact setting, for a large family of transitive countable Markov shifts, the space of invariant sub-probability measures is a Poulsen simplex and that its extreme points are the ergodic invariant probability measures together with the zero measure. In particular we obtain that the space of invariant probability measures is a Poulsen simplex minus a vertex and the corresponding convex combinations. Our results apply to finite…
We show that all non-trivial continuous endomorphisms of the circle group are topologically mixing. We also show that there exists a large infinite class of continuous endomorphisms of any n-dimensional torus group which are topologically mixing. Lastly, we prove that any continuous endomorphism on an abelian polish semigroup (with an identity) can be extended in a natural way to a topologically mixing endomorphism on the countable infinite product of said semigroup. This shows that every countable infinite product of an abelian polish semigroup has a topologically mixing endomorphism and, in …
In this paper, we study transitive partially hyperbolic diffeomorphisms with one-dimensional topologically neutral center, meaning that the length of the iterate of small center segments remains small. Such systems are dynamically coherent. We show that there exists a continuous metric along the center foliation which is invariant under the dynamics. As an application, we classify the transitive partially hyperbolic diffeomorphisms on 3-manifolds with topologically neutral center.
We construct a family of partially hyperbolic skew-product diffeomorphisms on $\mathbb{T}^3$ that are robustly transitive and admitting two physical measures with intermingled basins. In particularly, all these diffeomorphisms are not topologically mixing. Moreover, for every such example, it exhibits a dichotomy under perturbation: every perturbation of such example either has a unique physical measure and is robustly topologically mixing, or has two physical measures with intermingled basins.
For some countable discrete torsion Abelian groups we give examples of their finite measure-preserving actions which have simple spectrum and no approximate transitivity property.
We prove the existence of calibrated uniformly continuous subactions for coercive potentials with bounded variation defined on topologically transitive Markov shifts with countable alphabet through the construction of the Peierls barrier in this context. Also, we characterize the existence of bounded calibrated subactions in the same context.
We investigate the properties of chain recurrent, chain transitive, and chain mixing maps (generalizations of the well-known notions of non-wandering, topologically transitive, and topologically mixing maps). We describe the structure of chain transitive maps. These notions of recurrence are defined using $\ep$-chains, and the minimal lengths of these $\ep$-chains give a way to measure recurrence time (chain recurrence and chain mixing times). We give upper and lower bounds for these recurrence times and relate the chain mixing time to topological entropy.
We provide a quantification of the uniqueness of Gibbs measure for topologically mixing countable Markov shifts with locally Hölder continuous potentials. Corollaries for speed of convergence for approximation by finite subsystems are also given.
In this paper we generalize the recently introduced concept of fair measure (M. Misiurewicz and A. Rodrigues, Counting preimages. Ergod. Th. & Dynam. Sys. 38 (2018), no. 5, 1837 -- 1856). We study transitive countable state Markov shift maps and extend our results to a particular class of interval maps, Markov and mixing interval maps. Finally, we move beyond the interval and look for fair measures for graph maps.
Hoehn and Mouron [Ergod. Th. \& Dynam. Sys. (2014) \textbf{34}, 1897--1913] constructed a map on the universal dendrite that is topologically weakly mixing but not mixing. We modify the Hoehn-Mouron example to show that there exists a transitive (even weakly mixing) dendrite map with zero topological entropy. This answers the question of Baldwin [Topology (2001) \textbf{40}, 551--569].
On topologically transitive subsets of flag manifolds
Tata Institute of Fundamental Research（塔塔基础研究所）；Mathematisches Institut der Universität Bonn（波恩大学数学研究所）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学促进会数学研究所）
This paper shows that (1) there exists a topologically transitive NADS having two disjoint invariant periodic orbits with dense periodic points, which is finitely generated but not periodic; (2) there exists a topologically transitive non-finitely generated NADS having two disjoint invariant periodic orbits with dense periodic point, which is not sensitive. This answers positively the Open Problems 4.1 and 4.2 posed in [8].
We consider topological Markov chains (also called Markov shifts) on countable graphs. We show that a transient graph can be extended to a recurrent graph of equal entropy which is either positive recurrent of null recurrent, and we give an example of each type. We extend the notion of local entropy to topological Markov chains and prove that a transitive Markov chain admits a measure of maximal entropy (or maximal measure) whenever its local entropy is less than its (global) entropy.
In 1946, M. Morse proposed a conjecture that an analytic topologically transitive systems is metrically transitive. We prove this Morse conjecture for flows on a closed orientable surface of negative Euler characteristic. As a consequence, the Morse conjecture is true for highly transitive flows on non-orientable surfaces.
We study the equilibrium behaviour of a two-sided topological Markov shift with a countable number of states. We assume the potential associated with this shift is Walters with finite first variation and that the shift is topologically transitive. We show the equilibrium measure of the system is Bernoulli up to a period. In the process we generalize several theorems on countable Markov shifts. We prove a variational principle and the uniqueness of equilibrium measures. A key step is to show that functions with Walters property on a two-sided shift are cohomologous to one-sided functions with t…
We show that the existence of a dense set of periodic points for a topologically transitive non-minimal continuous group action on a Hausdorff uniform space with an infinite acting group does not necessarily imply a sensitive dependence to the initial conditions in such a system. This leads to define the chaos in the sense of Devaney for a continuous group action on a Hausdorff uniform spaces with an infinite acting group in the original way, i.e. a non-minimal topologically transitive and sensitive system with a dense set of periodic points is a chaotic system in the sense of Devaney.
Motivated from the study of multiple ergodic average, the investigation of multiplicative shift spaces has drawn much of interest among researchers. This paper focuses on the relation of topologically mixing properties between multiplicative shift spaces and traditional shift spaces. Suppose that $\mathsf{X}_Ω^{(l)}$ is the multiplicative subshift derived from the shift space $Ω$ with given $l > 1$. We show that $\mathsf{X}_Ω^{(l)}$ is (topologically) transitive/mixing if and only if $Ω$ is extensible/mixing. After introducing $l$-directional mixing property, we derive the equivalence between …
Within a subclass of monoids (with zero) a structural characterization is given of those that are associated to topologically transitive subshifts with Property (A).
We prove that every topologically transitive shift of finite type in one dimension is topologically conjugate to a subshift arising from a primitive random substitution on a finite alphabet. As a result, we show that the set of values of topological entropy which can be attained by random substitution subshifts contains all Perron numbers and so is dense in the positive real numbers. We also provide an independent proof of this density statement using elementary methods.
We consider random perturbations of a topologically transitive local diffeomorphism of a Riemannian manifold. We show that if an absolutely continuous ergodic stationary measures is expanding (all Lyapunov exponents positive), then there is a random Gibbs-Markov-Young structure which can be used to lift that measure. We also prove that if the original map admits a finite number of expanding invariant measures then the stationary measures of a sufficiently small stochastic perturbation are expanding.
