Jana Hantáková

• RNDr.
• Researcher at Silesian University in Opava

Metrizable spaces are studied in which every closed set is an α-limit set for some continuous map and some point. It is shown that this property is enjoyed by every space containing sufficiently many arcs (formalized in the notion of a space with enough arcs), though such a space need not be arcwise connected. Further it is shown that this property...

We are interested in dendrites for which all invariant measures of zero-entropy mappings have discrete spectrum, and we prove that this holds when the closure of the endpoint set of the dendrites is countable. This solves an open question which has been around for awhile, and almost completes the characterization of dendrites with this property.

Metrizable spaces are studied in which every closed set is an $\alpha$-limit set for some continuous map and some point. It is shown that this property is enjoyed by every space containing sufficiently many arcs (formalized in the notion of a space with enough arcs), though such a space need not be arcwise connected. Further it is shown that this p...

Special α -limit sets ( sα -limit sets) combine together all accumulation points of all backward orbit branches of a point x under a noninvertible map. The most important question about them is whether or not they are closed. We challenge the notion of sα -limit sets as backward attractors for interval maps by showing that they need not be closed....

In the paper we study what sets can be obtained as $\alpha$-limit sets of backward trajectories in graph maps. We show that in the case of mixing maps, all those $\alpha$-limit sets are $\omega$-limit sets and for all but finitely many points $x$, we can obtain every $\omega$-limits set as the $\alpha$-limit set of a backward trajectory starting in...

We are interested in dendrites for which all invariant measures of zero-entropy mappings have discrete spectrum, and we prove that this holds when the closure of the endpoint set of the dendrite is countable. This solves an open question which was around for awhile, almost completing the characterization of dendrites with this property.

In the paper we study what sets can be obtained as \begin{document}$ \alpha $\end{document}-limit sets of backward trajectories in graph maps. We show that in the case of mixing maps, all those \begin{document}$ \alpha $\end{document}-limit sets are \begin{document}$ \omega $\end{document}-limit sets and for all but finitely many points \begin{docu...

Special $\alpha$-limit sets ($s\alpha$-limit sets) combine together all accumulation points of all backward orbit branches of a point $x$ under a noninvertible map. The most important question about them is whether or not they are closed. We challenge the notion of $s\alpha$-limit sets as backward attractors for interval maps by showing that they n...

We construct an infinite-dimensional compact metric space $X$, which is a closed subset of $\mathbb{S}\times\mathbb{H}$, where $\mathbb{S}$ is the unit circle and $\mathbb{H}$ is the Hilbert cube, and a skew-product map $F$ acting on $X$ such that $(X,F)$ is Li-Yorke sensitive but possesses at most countable scrambled sets. This disproves the conje...

We construct an infinite-dimensional compact metric space $X$, which is a closed subset of $\mathbb{S}\times\mathbb{H}$, where $\mathbb{S}$ is the unit circle and $\mathbb{H}$ is the Hilbert cube, and a skew-product map $F$ acting on $X$ such that $(X,F)$ is Li-Yorke sensitive but possesses at most countable scrambled sets. This disproves the conje...

We disprove the conjecture that the existence of a DC3-scrambled pair is preserved under iteration and show that a slightly strengthened definition of distributional chaos of type 3, denoted by DC21 2, is iteration invariant, i.e. that fn is DC21 2 if and only if f is. Unlike DC3, DC21 2 is also conjugacy invariant and implies Li–Yorke chaos. The d...

The aim of the paper is to correct and improve some results concerning distributional chaos of type 3. We show that in a general compact metric space, distributional chaos of type 3, denoted DC3, even when assuming the existence of an uncountable scrambled set, is a very weak form of chaos. In particular, (i) the chaos can be unstable (it can be de...

We disprove the conjecture that distributional chaos of type 3 (briefly, DC3) is iteration invariant and show that a slightly strengthened definition, denoted by DC2$\frac{1}{2}$, is preserved under iteration, i.e. $f^n$ is DC2$\frac{1}{2}$ if and only if $f$ is too. Unlike DC3, DC2$\frac{1}{2}$ is also conjugacy invariant and implies Li-Yorke chao...

We show that in a general compact metric space, distributional chaos of type 3, denoted DC3, even when assuming existence of an uncountable scrambled set, is a very weak form of chaos. In particular, (i) the chaos can be unstable (it can be destroyed by conjugacy), and (ii) such an unstable system may contain no Li-Yorke pair. However, definition c...

We show the existence of a dynamical system without any distributionally scrambled pair which is semiconjugated to a distributionally chaotic factor.

This article investigates the relation between the distributional chaos and the existence of a scrambled triple. We show that for a continuous mapping $f$ acting on a compact metric space $(X,d)$, the possession of an infinite extremal distributionally scrambled set is not sufficient for the existence of a scrambled triple. We also construct an inv...

The paper solves a question posed by Oprocha on the existence of invariant distributionally chaotic scrambled sets. We show, among other things, that a continuous map ff acting on compact metric space (X,d)(X,d) with a weak specification property, fixed point, and infinitely many mutually distinct periods has a dense Mycielski (i.e., cc dense set o...