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DCDS

Let G ↷ X be a topological action of a topological semigroup $G$ on a compact metric space $X$. We show in this paper that for any given point $x$ in $X$, the following two properties that both approximate to periodicity are equivalent to each other:

$\bullet$ For any neighborhood $U$ of $x$, the return times set $\{g\in G ： gx\in U\}$ is syndetic of Furstenburg in $G$.

$\bullet$ Given any $\varepsilon>0$, there exists a finite subset $K$ of $G$ such that for each $g$ in $G$, the $\varepsilon$-neighborhood of the orbit-arc $K[gx]$ contains the entire orbit $G[x]$.

This is a generalization of a classical theorem of Birkhoff for the case where $G=\mathbb{R}$ or $\mathbb{Z}$. In addition, a counterexample is constructed to this statement, while $X$ is merely a complete but not locally compact metric space.

$\bullet$ For any neighborhood $U$ of $x$, the return times set $\{g\in G ： gx\in U\}$ is syndetic of Furstenburg in $G$.

$\bullet$ Given any $\varepsilon>0$, there exists a finite subset $K$ of $G$ such that for each $g$ in $G$, the $\varepsilon$-neighborhood of the orbit-arc $K[gx]$ contains the entire orbit $G[x]$.

This is a generalization of a classical theorem of Birkhoff for the case where $G=\mathbb{R}$ or $\mathbb{Z}$. In addition, a counterexample is constructed to this statement, while $X$ is merely a complete but not locally compact metric space.

DCDS

In this note, we improve a combinatorial sifting-type lemma obtained
in [11].More precisely, we sift out a continuous infinite
"$(\xi_1,\xi_2)$-Liao string" sequence for any real sequence
$\{a_i\}_1^\infty$ with
$\limsup_{n\to\infty}{n}^{-1}\sum_{i=1}^na_i=\xi\in(\xi_1,\xi_2)$.

DCDS

Let $X$ be a separable metric space not necessarily compact, and let
$f: X\rightarrow X$ be a continuous transformation. From the
viewpoint of Hausdorff dimension, the authors improve Bowen's method
to introduce a dynamical quantity distance entropy, written as
$ent_{H}(f;Y)$, for $f$ restricted on any given subset $Y$
of $X$; but it is essentially different from Bowen's entropy(1973).
This quantity has some basic properties similar to Hausdorff
dimension and is beneficial to estimating Hausdorff dimension of the
dynamical system. The authors show that if $f$ is a local
lipschitzian map with a lipschitzian constant $l$ then
$ent_{H}(f;Y)\le\max\{0, $HD$(Y)\log l}$ for all
$Y\subset X$; if $f$ is locally expanding with skewness $\lambda$
then $ent_{H}(f;Y)\ge $HD$(Y)\log\lambda$ for any
$Y\subset X$. Here HD$(-)$ denotes the Hausdorff dimension.
The countable stability of the distance entropy $ent_{H}$
proved in this paper, which generalizes the finite stability of
Bowen's $h$-entropy (1971), implies that a continuous pointwise
periodic map has the distance entropy zero. In addition, the authors
show examples which demonstrate that this entropy describes the real
complexity for dynamical systems over noncompact-phase space better
than that of various other entropies.

ERA-MS

Based on the classic multiplicative ergodic theorem and the semi-uniform subadditive ergodic theorem, we show that there always exists at least one ergodic Borel probability measure such that
the joint spectral radius of a finite set of square matrices of the same size can be realized almost everywhere with respect to this Borel probability measure. The existence of at least one ergodic Borel probability measure, in the context of the joint spectral radius problem, is obtained in a general setting.

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