arXiv subjects

Jana Hantáková

Publications and source records attributed to Jana Hantáková.

8 recordsLinked to original sources

arXiv2026

Observable Dynamics and the Generic Coincidence of Milnor, Statistical, and Physical Attractors

We study observable dynamics of generic continuous interval maps. We prove that, for a residual subset of $C([0,1])$, the global Milnor, statistical, and physical attractors coincide with the non-wandering set. Thus, for a typical interval map, the asymptotic dynamics detected by Lebesgue-almost every initial condition recovers all recurrent dynamics. The common attractor is a robust Cantor set of zero Hausdorff dimension. Moreover, every invariant probability measure has a basin of zero Lebesgue measure. We also present examples showing that, outside the generic setting, positive topological entropy may occur outside the global Milnor attractor, and the three attractors need not coincide.

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arXiv2022

Spaces where all closed sets are $α$-limit sets

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 is not preserved by topological sums, products and continuous images and quotients. However, positive results do hold for metrizable spaces obtained by those constructions from spaces with enough arcs.

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arXiv2021

On the structure of $α$-limit sets of backward trajectories for graph maps

In the paper we study what sets can be obtained as $α$-limit sets of backward trajectories in graph maps. We show that in the case of mixing maps, all those $α$-limit sets are $ω$-limit sets and for all but finitely many points $x$, we can obtain every $ω$-limits set as the $α$-limit set of a backward trajectory starting in $x$. For zero entropy maps, every $α$-limit set of a backward trajectory is a minimal set. In the case of maps with positive entropy, we obtain a partial characterization which is very close to complete picture of the possible situations.

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arXiv2021

Dendrites and measures with discrete spectrum

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.

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arXiv2020

On backward attractors of interval maps

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. This disproves a conjecture by Kolyada, Misiurewicz, and Snoha. We give a criterion in terms of Xiong's attracting center that completely characterizes which interval maps have all $sα$-limit sets closed, and we show that our criterion is satisfied in the piecewise monotone case. We apply Blokh's models of solenoidal and basic $ω$-limit sets to solve four additional conjectures by Kolyada, Misiurewicz, and Snoha relating topological properties of $sα$-limit sets to the dynamics within them. For example, we show that the isolated points in a $sα$-limit set of an interval map are always periodic, the non-degenerate components are the union of one or two transitive cycles of intervals, and the rest of the $sα$-limit set is nowhere dense. Moreover, we show that $sα$-limit sets in the interval are always both $F_σ$ and $G_δ$. Finally, since $sα$-limit sets need not be closed, we propose a new notion of $β$-limit sets to serve as backward attractors. The $β$-limit set of $x$ is the smallest closed set to which all backward orbit branches of $x$ converge, and it coincides with the closure of the $sα$-limit set. At the end of the paper we suggest several new problems about backward attractors.

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arXiv2017

Li-Yorke sensitivity does not imply Li-Yorke chaos

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 conjecture of Akin and Kolyada that Li-Yorke sensitivity implies Li-Yorke chaos from the article [Akin E., Kolyada S., Li-Yorke sensitivity, Nonlinearity 16, (2003), 1421-1433].

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arXiv2016

Iteration problem for distributional chaos

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 chaos. The definition of DC2$\frac{1}{2}$ is the following: a pair $(x,y)$ is DC2$\frac{1}{2}$ iff $Φ_{(x,y)}(0)<Φ^*_{(x,y)}(0)$, where $Φ_{(x,y)}(δ)$ (resp. $Φ^*_{(x,y)}(δ)$) is lower (resp. upper) density of times $k$ when $d(f^k(x),f^k(y))<δ$ and both densities are defined at 0 as limits of their values for $δ\to 0^+$. Hence DC$2\frac{1}{2}$ shares similar properties with DC1 and DC2 but unlike them, strict DC$2\frac{1}{2}$ systems must have zero topological entropy.

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arXiv2016

On the weakest version of distributional chaos

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 destroyed by conjugacy), and (ii) such an unstable system may contain no Li-Yorke pair. However, the definition can be strengthened to get DC$2\frac{1}{2}$ which is a topological invariant and implies Li-Yorke chaos, similarly as types DC1 and DC2; but unlike them, strict DC$2\frac{1}{2}$ systems must have zero topological entropy.

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