We consider dynamical systems of the form $(X,f)$ where $X$ is a compact metric space and $f:X\to X$ is either a continuous map or a homeomorphism and provide a new proof that there is no universal metric dynamical system of this kind. The same is true for metric minimal dynamical systems and for metric abstract $\omega$-limit sets, answering a question by Will Brian.

We consider binary rotation words generated by partitions of the unit circle to two intervals and give a precise formula for the number of such words of length n. We also give the precise asymptotics for it, which happens to be Θ(n4). The result continues the line initiated by the formula for the number of all Sturmian words obtained by Lipatov [Problemy Kibernet. 39 (1982) 67–84], then independently by Mignosi [Theoret. Comput. Sci. 82 (1991) 71–84], and others.

In this paper, we discuss the properties of limit sets of subsets and attractors in a compact metric space. It is shown that the $\omega$-limit set $\omega \left(Y\right)$ of $Y$ is the limit point of the sequence ${\{\left(\mathrm{C}lY\right)·[i,\infty )\}}_{i=1}^{\infty }$ in $2^X$ and also a quasi-attractor is the limit point of attractors with respect to the Hausdorff metric. It is shown that if a component of an attractor is not an attractor, then it must be a real quasi-attractor.

It is shown that each expansive group action on a Peano continuum having a free dendrite must have a ping-pong game, and has positive geometric entropy when the acting group is finitely generated. As a corollary, it is shown that each Peano continuum having a free dendrite admits no expansive nilpotent group actions.

We investigate a weighted version of Hausdorff dimension introduced by V. Afraimovich, where the weights are determined by recurrence times. We do this for an ergodic invariant measure with positive entropy of a piecewise monotonic transformation on the interval $[0,1]$, giving first a local result and proving then a formula for the dimension of the measure in terms of entropy and characteristic exponent. This is later used to give a relation between the dimension of a closed invariant subset and a pressure...

We discuss the existence of an uncountable strongly chaotic set of a continuous self-map on a compact metric space. It is proved that if a continuous self-map on a compact metric space has a regular shift invariant set then it has an uncountable strongly chaotic set in which each point is recurrent, but is not almost periodic.

For n ≥ 1, given an n-dimensional locally (n-1)-connected compact space X and a finite Borel measure μ without atoms at isolated points, we prove that for a generic (in the uniform metric) continuous map f:X → X, the set of points which are chain recurrent under f has μ-measure zero. The same is true for n = 0 (skipping the local connectedness assumption).

The aim of the paper is to investigate the structure of disjoint iteration groups on the unit circle ${𝕊}^1$, that is, families $ℱ=\{F^v{𝕊}^1⟶{𝕊}^{1}v\in V\}$ of homeomorphisms such that $$F^{v_1}\circ F^{v_2}=F^{v_1+v_2},v_1,v_2\in V,$$ and each $F^v$ either is the identity mapping or has no fixed point ($(V,+)$ is an arbitrary $2$-divisible nontrivial (i.e., $\mathrm{c}ardV>1$) abelian group).

We answer affirmatively Coven's question [PC]: Suppose f: I → I is a continuous function of the interval such that every point has at least two preimages. Is it true that the topological entropy of f is greater than or equal to log 2?

Let f: [a,b] → [a,b] be a continuous function of the compact real interval such that (i) $card{f}^{-1}\left(y\right)\ge 2$ for every y ∈ [a,b]; (ii) for some m ∈ ∞,2,3,... there is a countable set L ⊂ [a,b] such that $card{f}^{-1}\left(y\right)\ge m$ for every y ∈ [a,b]∖L. We show that the topological entropy of f is greater than or equal to log m. This generalizes our previous result for m = 2.

Each topological group G admits a unique universal minimal dynamical system (M(G),G). For a locally compact noncompact group this is a nonmetrizable system with a rich structure, on which G acts effectively. However there are topological groups for which M(G) is the trivial one-point system (extremely amenable groups), as well as topological groups G for which M(G) is a metrizable space and for which one has an explicit description. We show that for the topological group G = Homeo(E) of self-homeomorphisms...