Michal Málek | Searcharxiv

arXiv2025

Topological Transitivity of Nonautonomous Dynamical Systems

This paper explores the concept of topological transitivity in nonautonomous dynamical systems, which are defined as sequences of continuous maps from a compact metric space to itself. It investigates various conditions (including intersection of any pair of open sets and existence of a dense orbit) that could be taken as definitions of the topological transitivity of a nonautonomous system, and addresses their relation both in the case of a general compact metric space and in the case where, in addition, the space has no isolated point. This provides the necessary basis for further investigation of transitivity of nonautonomous dynamical systems.

math.DS↗

arXiv2015

Omega-limit sets and invariant chaos in dimension one

Omega-limit sets play an important role in one-dimensional dynamics. During last fifty year at least three definitions of basic set has appeared. Authors often use results with different definition. Here we fill in the gap of missing proof of equivalency of these definitions. Using results on basic sets we generalize results in paper [P.~Oprocha, Invariant scrambled sets and distributional chaos, Dyn. Syst. 24 (2009), no. 1, 31--43.] to the case continuous maps of finite graphs. The Li-Yorke chaos is weaker than positive topological entropy. The equivalency arises when we add condition of invariance to Li-Yorke scrambled set. In this note we show that for a continuous graph map properties positive topological entropy; horseshoe; invariant Li-Yorke scrambled set; uniform invariant distributional chaotic scrambled set and distributionaly chaotic pair are mutually equivalent.

math.DS↗