arXiv subjects

Michaela Mlíchová

Publications and source records attributed to Michaela Mlíchová.

3 recordsLinked to original sources

arXiv2021

Transitivity in nonautonomous systems

Let $(X,d)$ be a metric space and $f_{1,\infty}=\{f_n\}_{i=0}^{\infty}$ be a sequence of continuous maps $f_n:X\rightarrow X$ such that $(f_n)$ converges uniformly to a continuous map $f$. We investigate which conditions ensure that the transitivity of functions $f_n$ or the transitivity of the nonautonomous system $(X,f_{1,\infty})$ is inherited to the limit function $f$ and vice versa. Such problem have been studied for instance by A. Fedeli, A. Le Donne or J. Li who give different sufficient condition for inheriting of transitivity from $f_n$ to $f$. In this paper we give a survey of know result relating to this problem and prove new results concerning to transitivity.

math.DS↗

arXiv2016

Li-Yorke sensitive and weak mixing dynamical systems

Akin and Kolyada in 2003 [E. Akin, S. Kolyada, Li-Yorke sensitivity, Nonlinearity 16 (2003) 1421 - 1433] introduced the notion of Li-Yorke sensitivity. They proved that every weak mixing system $(X, T)$, where $X$ is a compact metric space and $T$ a continuous map of $X$ is Li-Yorke sensitive. An example of Li-Yorke sensitive system without weak mixing factors was given in [M. Čiklová, Li-Yorke sensitive minimal maps, Nonlinearity 19 (2006) 517 - 529] (see also [M. Čiklová-Ml\'ıchová, Li-Yorke sensitive minimal maps II, Nonlinearity 22 (2009) 1569 -1573]). In their paper, Akin and Kolyada conjectured that every minimal system with a weak mixing factor, is Li-Yorke sensitive. We provide arguments supporting this conjecture though the proof seems to be difficult.

math.DS↗

arXiv2016

On generic and dense chaos for maps induced on hyperspaces

A continuous map $f$ on a compact metric space $X$ induces in a natural way the map $\tilde f$ on the hyperspace $\mathcal K(X)$ of all closed non-empty subsets of $X$. We study the question of transmission of chaos between $f$ and $\tilde f$. We deal with generic, generic $\varepsilon$-, dense and dense $\varepsilon$-chaos for interval maps. We prove that all four types of chaos transmit from $f$ to $\tilde f$, while the converse transmission from $\tilde f$ to $f$ is true for generic, generic $\varepsilon$- and dense $\varepsilon$-chaos. Moreover, the transmission of dense $\varepsilon$- and generic $\varepsilon$-chaos from $\tilde f$ to $f$ is true for maps on general compact metric spaces.

math.DS↗