Dominik Kwietniak

On almost specification and average shadowing properties

In this paper we study relations between almost specification property, asymptotic average shadowing property and average shadowing property for dynamical systems on compact metric spaces. We show implications between these properties and relate them to other important notions such as shadowing, transitivity, invariant measures, etc. We provide examples that compactness is a necessary condition for these implications to hold. As a consequence of our methodology we also obtain a proof that limit shadowing in chain transitive systems implies shadowing.

Exact Devaney chaos and entropy

We modify the definition of chaos in the sense of Devaney, by replacing the condition of topological transitivity by topological exactness. We study basic properties of exact Devaney chaos defined in such a way. We also investigate the infimum of topological entropies of exactly Devaney chaotic maps of a given space.

A note on transitivity, sensitivity and chaos for graph maps

Two elementary proofs showing that (i) transitivity and sensitivity imply dense periodicity for maps on topological graphs and (ii) total transitivity and dense periodicity imply mixing for maps on spaces with an open subset homeomorphic with the open interval (0,1) are presented. As corollaries, one gets new and simple proofs that Auslander–Yorke chaos implies Devaney chaos, and weak mixing implies mixing for graph maps.

When lower entropy implies stronger Devaney chaos

It is proved that the infimum of the topological entropy of continuous topologically exact interval (circle) maps is strictly smaller than the infimum of the topological entropy of continuous interval (circle) maps, which are topologically mixing, but not exact. Interpreting this result in terms of popular notions of chaos, one may say that on the interval (circle) lower entropy implies stronger Devaney chaos. Moreover, the infimum of the entropy of mixing circle maps is computed. These theorems may be considered as a completion of some results of Alseda, Kolyada, Llibre, and Snoha (1999).

Multi-recurrence and van der Waerden systems

We explore recurrence properties arising from dynamical approach to the van der Waerden theorem and similar combinatorial problems. We describe relations between these properties and study their consequences for dynamics. In particular, we present a measure-theoretical analog of a result of Glasner on multi-transitivity of topologically weakly mixing minimal maps. We also obtain a dynamical proof of the existence of a C-set with zero Banach density.

Generic points for dynamical systems with average shadowing

We study the Besicovitch pseudometric

Shadowing vs. distality for actions of ℝ n

D_B

Transitive dendrite map with zero entropy

DB for compact dynamical systems. The set of generic points of ergodic measures turns out to be closed with respect to