arXiv subjects

Magdalena Foryś-Krawiec

Publications and source records attributed to Magdalena Foryś-Krawiec.

7 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.

math.DS↗

arXiv2024

Metric mean dimension of irregular sets for maps with shadowing

We study the metric mean dimension of $Φ$-irregular set $I_Φ(f)$ in dynamical systems with the shadowing property. In particular we prove that for dynamical systems with shadowing containing a chain recurrent class $Y$, the values of topological entropy together with the values of lower and upper metric mean dimension of the set $I_Φ(f)\cap B(Y,\varepsilon)\cap CR(f)$ are bounded from below by the respective values for class $Y$.

math.DS↗

arXiv2023

A note on home-product-minimality

A compact space $Y$ is called homeo-product-minimal if given any minimal system $(X,T)$, it admits a homeomorphism $S:Y\to Y$, such that the product system $(X\times Y,T\times S)$ is minimal. We show that a large class of cofrontiers is homeo-product-minimal. This class contains R. H. Bing's pseudo-circle, answering a question of Dirbák, Snoha and Špitalský from [Minimal direct products, Trans. Amer. Math. Soc. 375 (2022)].

math.DS↗

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.

math.DS↗

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.

math.DS↗

arXiv2020

On Rigid Minimal Spaces

A compact space $X$ is said to be minimal if there exists a map $f:X\to X$ such that the forward orbit of any point is dense in $X$. We consider rigid minimal spaces, motivated by recent results of Downarowicz, Snoha, and Tywoniuk [J. Dyn. Diff. Eq., 2016] on spaces with cyclic group of homeomorphisms generated by a minimal homeomorphism, and results of the first author, Clark and Oprocha [ Adv. Math., 2018] on spaces in which the square of every homeomorphism is a power of the same minimal homeomorphism. We show that the two classes do not coincide, which gives rise to a new class of spaces that admit minimal homeomorphisms, but no minimal maps. We modify the latter class of examples to show for the first time the existence of minimal spaces with degenerate homeomorphism groups. Finally, we give a method of constructing decomposable compact and connected spaces with cyclic group of homeomorphisms, generated by a minimal homeomorphism, answering a question in Downarowicz et al.

math.DS↗

arXiv2019

On entropy of $Φ$-irregular and $Φ$-level sets in maps with the shadowing property

We study the properties of $Φ$-irregular sets (sets of points for which the Birkhoff average diverges) in dynamical systems with the shadowing property. We estimate the topological entropy of $Φ$-irregular set in terms of entropy on chain recurrent classes and prove that $Φ$-irregular sets of full entropy are typical. We also consider $Φ$-level sets (sets of points whose Birkhoff average is in a specified interval), relating entropy they carry with the entropy of some ergodic measures. Finally, we study the problem of large deviations considering the level sets with respect to reference measures.

math.DS↗