Related papers: Measure-theoretic equicontinuity and rigidity

We study the structure of invariant measures for continuous automorphisms of compact metrizable abelian groups satisfying the descending chain condition. We show that the finitely supported invariant measures are weak-* dense in the space…

Let $M$ be a compact Riemannian manifold. The set $\text{F}^{r}(M)$ consisting of sequences $(f_{i})_{i\in\mathbb{Z}}$ of $C^{r}$-diffeomorphisms on $M$ can be endowed with the compact topology or with the strong topology. A notion of…

Let $\mathcal{M}(X,\mathcal{A},\mu)$ be the ring of all real-valued measurable functions constructed over a measure space $(X,\mathcal{A},\mu)$. A topology on $\mathcal{M}(X,\mathcal{A},\mu)$, called the {$F_\mu$-topology} weaker than the {…

For a fixed topological Markov shift, we consider measure-preserving dynamical systems of Gibbs measures for 2-locally constant functions on the shift. We also consider isomorphisms between two such systems. We study the set of all…

Let $(X,T)$ be a topological dynamical system with metric $d$. We define a new function $\overline{F}(x,y)=\limsup\limits_{n \to +\infty} \inf\limits_{\sigma \in S_n} \frac 1n \sum\limits_{k=1}^n d(T^k x,T^{\sigma(k)} y)$ by using…

In this paper we examine the interplay between recurrence properties and the shadowing property in dynamical systems on compact metric spaces. In particular, we demonstrate that if the dynamical system $(X,f)$ has shadowing, then it is…

Let $(X,G)$ be a topological dynamical system, given by the action of a is a countable discrete infinite group on a compact metric space $X$. We prove that if $(X,G)$ is minimal, then it is either diam-mean $m$-equicontinuious or diam-mean…

We say that a system of differential equations d^2x(t)/dt^2=Adx(t)/dt+Bx(t)+Cu(t), in which A and B are m-by-m complex matrices and C is an m-by-n complex matrix, is rigid if it can be reduced by substitutions x(t)=Sy(t),…

We consider flows $(X,T)$, given by actions $(t, x) \to tx$, on a compact metric space $X$ with a discrete $T$ as an acting group. We study a new class of flows - the \textsc{Strongly Rigid} ($ \mathbf {SR} $) \ flows, that are properly…

We prove the following entropy-rigidity result in finite volume: if $X$ is a negatively curved manifold with curvature $-b^2\leq K_X \leq -1$, then $Ent_{top}(X) = n-1$ if and only if $X$ is hyperbolic. In particular, if $X$ has the same…

We investigate the connections between independence, sequence entropy, and mean sensitivity for a measure preserving system under the action of a countable infinite discrete group. We establish that every sequence entropy tuple for an…

Stochastic dynamical systems consisting of non-invertible continuous maps on an interval are studied. It is proved that if they satisfy the recently introduced so-called $\mu$-injectivity and some mild assumptions, then proximality,…

We show that if a (locally compact) group $G$ acts properly on a locally compact $\sigma$-compact space $X$ then there is a family of $G$-invariant proper continuous finite-valued pseudometrics which induces the topology of $X$. If $X$ is…

Given a measurable dynamical system $(X,\mathcal{X},\mu,T)$, where $X$ is a compact metric space, $\mathcal{X}$ is the Borel $\sigma$-algebra on $X$, $\mu$ is a $T$-invariant Borel probability measure and $T$ is a homeomorphism acting on…

Let $X$ be a compact metrizable group and $\Gamma$ a countable group acting on $X$ by continuous group automorphisms. We give sufficient conditions under which the dynamical system $(X,\Gamma)$ is surjunctive, i.e., every injective…

We show that if $(X,T)$ is an extension of an aperiodic subshift (a subsystem of $({1,2,...,l}^{\mathbb{Z}},\mathrm{shift})$ for some $l\in\mathbb{N}$) and has mean dimension $mdim(X,T)<\frac{D}{2}$ $(D\in \mathbb{N}$), then it embeds…

We consider topological dynamical systems given by skew products $S\rtimes_{\tau} T$, where $S\colon Y\to Y$ is a subshift, $\tau\colon Y\to\mathbb{Z}$ is a continuous cocycle, and $T$ is an arbitrary invertible topological system. For…

Answering an open question affirmatively it is shown that every ergodic invariant measure of a mean equicontinuous (i.e. mean-L-stable) system has discrete spectrum. Dichotomy results related to mean equicontinuity and mean sensitivity are…

A probability measure preserving action of \Gamma on (X,\mu) is called rigid if the inclusion of L^\infty(X) into the crossed product L^\infty(X) \rtimes \Gamma has the relative property (T) in the sense of Popa. We give examples of rigid,…

This article is devoted to study which conditions imply that a topological dynamical system is mean sensitive and which do not. Among other things we show that every uniquely ergodic, mixing system with positive entropy is mean sensitive.…