Related papers: Dynamics on Hyperspace of Pointwise Periodic Homeo…

Let $f:X\to X$ be a self homeomorphism of a continuum $X$, we show that the topological entropy of the induced system $(2^X,2^f)$ is infinite provided that $X\setminus \Omega(f)$ is not empty. If furthermore $X$ is a regular curve then it…

In this paper, we prove first that the space of minimal sets of any homeomorphisms $f:X\to X$ of a regular curve $X$ is closed in the hyperspace $2^X$ of closed subsets of $X$ endowed with the Hausdorff metric, and the non-wandering set…

Given a self-map of a compact metric space $X$, we study periodic points of the map induced on the hyperspace of closed subsets of $X$. We give some necessary conditions on admissible sets of periods for these maps. Seemingly unrelated to…

In this paper we prove that every homeomorphism of a compact metric space admitting an invariant probability measure with full support can be approximated in the $C^0$-Gromov--Hausdorff topology by homeomorphisms with zero topological…

We use recent developments in local entropy theory to prove that chaos in dynamical systems implies the existence of complicated structure in the underlying space. Earlier Mouron proved that if $X$ is an arc-like continuum which admits a…

Let $f : M \rightarrow M$ be a Morse-Smale diffeomorphism defined on a compact and connected manifold without boundary. Let $C(M)$ denote the hyperspace of all subcontinua of M endowed with the Hausdorff metric and $C(f) : C(M) \rightarrow…

We study a class of asymptotically entropy-expansive $C^1$ diffeomorphisms with dominated splitting on a compact manifold $M$, that satisfy the specification property. This class includes, in particular, transitive Anosov diffeomorphisms…

We study the dynamics of continuous maps on compact metric spaces containing a free interval (an open subset homeomorphic to the interval $(0,1)$). We provide a new proof of a result of M. Dirb\'ak, \v{L}. Snoha, V. \v{S}pitalsk\'y [Ergodic…

In this paper, we study the dynamical properties of actions on the space of compact subsets of the phase space. More precisely, if $X$ is a metric space, let $2^X$ denote the space of non-empty compact subsets of $X$ provided with the…

We prove that every $C^\infty$-smooth, area preserving diffeomorphism of the closed 2-disk having not more than one periodic point is the uniform limit of periodic $C^\infty$-smooth diffeomorphisms. In particular every smooth irrational…

A homeomorphism of a compact metric space is {\em tight} provided every non-degenerate compact connected (not necessarily invariant) subset carries positive entropy. It is shown that every $C^{1+\alpha}$ diffeomorphism of a closed surface…

We prove that local stable/unstable sets of homeomorphisms of an infinite compact metric space satisfying the gluing-orbit property always contain compact and perfect subsets of the space. As a consequence, we prove that if a positively…

In this work, we show that if $f$ is a uniformly continuous map defined over a Polish metric space, then the set of $f$-invariant measures with zero metric entropy is a $G_\delta$ set (in the weak topology). In particular, this set is…

In this note we study some properties of topological entropy for non-compact non-metrizable spaces. We prove that if a uniformly continuous self-map $f$ of a uniform space has topological shadowing property then the map $f$ has positive…

In this paper we study interval maps $f$ with zero topological entropy that are crooked; i.e. whose inverse limit with $f$ as the single bonding map is the pseudo-arc. We show that there are uncountably many pairwise non-conjugate zero…

Let $X$ be a dendrite with set of endpoints $E(X)$ closed and let $f:~X \to X$ be a continuous map with zero topological entropy. Let $P(f)$ be the set of periodic points of $f$. We prove that if $L$ is an infinite $\omega$-limit set of $f$…

We show that if $G$ is a topological graph, and $f$ is continuous map, then the induced map $\tilde{f}$ acting on the hyperspace $C(G)$ of all connected subsets of $G$ by natural formula $\tilde{f}(C)=f(C)$ carries the same entropy as $f$.…

We prove that self-mappings of uniquely arcwise connected locally arcwise connected spaces are pointwise-recurrent if and only if all their cutpoints are periodic while all endpoints are either periodic or belong to what we call…

We show that in any family of stunted sawtooth maps, the set of maps whose set of periods is the set of all powers of 2 has no interior point, i.e., the combinatorial description of the boundary of chaos coincides with the topological…

A continuous map $f$ from a compact interval $I$ into itself is densely (resp. generically) chaotic if the set of points $(x,y)$ such that $\limsup_{n\to+\infty}|f^n(x)-f^n(y)|>0$ and $\liminf_{n\to+\infty} |f^n(x)-f^n(y)|=0$ is dense…