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Ercai Chen

Publications and source records attributed to Ercai Chen.

At least 55 records · Page 3Linked to original sources
arXiv2018

On the topological entropy of the set with a special shadowing time

Let $(X,d,T )$ be a topological dynamical system with specification property. For $ α\in \mathbb R^+$ and any $x_0\in X$, define $$ \mathbf D^{x_0}_α:=\Big\{x\in X: \lim\limits_{ε\to 0}\limsup\limits_{n\to\infty}\dfrac{\max\{t\in\mathbb N:~T^n(x)\in B_{t}( x_0,ε)\}}{n}\geq α\Big\}.$$ Then we have $h_{top}^{B} ( T, \mathbf D^{x_0}_α)= \dfrac{h_{top}(T)}{1+α} $, where $h_{top}^{B} ( T, \mathbf D^{x_0}_α)$ denotes the Bowen topological entropy of $\mathbf D^{x_0}_α.$

math.DS↗
arXiv2017

Topological entropy of sets of generic points for actions of amenable groups

Let $G$ be a countable discrete amenable group which acts continuously on a compact metric space $X$ and let $μ$ be an ergodic $G-$invariant Borel probability measure on $X$. For a fixed tempered Følner sequence $\{F_n\}$ in $G$ with $\lim\limits_{n\rightarrow+\infty}\frac{|F_n|}{\log n}=\infty$, we prove the following variational principle: $$h^B(G_μ,\{F_n\})=h_μ(X,G),$$ where $G_μ$ is the set of generic points for $μ$ with respect to $\{F_n\}$ and $h^B(G_μ,\{F_n\})$ is the Bowen topological entropy (along $\{F_n\}$) on $G_μ$. This generalizes the classical result of Bowen in 1973.

math.DS↗
arXiv2016

The Conditional Variational Principle for Maps with the Pseudo-orbit Tracing Property

Let $(X,d,f)$ be a topological dynamical system, where $(X,d)$ is a compact metric space and $f:X\to X$ is a continuous map. We define $n$-ordered empirical measure of $x\in X$ by \begin{align*} \mathscr{E}_n(x)=\frac{1}{n}\sum\limits_{i=0}^{n-1}δ_{f^ix}, \end{align*} where $δ_y$ is the Dirac mass at $y$. Denote by $V(x)$ the set of limit measures of the sequence of measures $\mathscr{E}_n(x).$ In this paper, we obtain conditional variational principles for the topological entropy of \begin{align*} Δ_{sub}(I)=\left\{x\in X:V(x)\subset I\right\}, \end{align*} and \begin{align*} Δ_{cap}(I)=\left\{x\in X:V(x)\cap I\neq\emptyset \right\}. \end{align*} in a transitive dynamical system with the pseudo-orbit tracing property, where $I$ is a certain subset of $\mathscr M_{\rm inv}(X,f)$.

math.DS↗
arXiv2016

Mean Topological Dimension for random bundle transformations

We introduce the mean topological dimension for random bundle transformations, and show that continuous bundle random dynamical systems with finite topological entropy, or the small boundary property have zero mean topological dimensions.

math.DS↗
arXiv2016

Bowen entropy for actions of amenable groups

Bowen introduced a definition of topological entropy of subset inspired by Hausdorff dimension in 1973 \cite{B}. In this paper we consider the Bowen's entropy for amenable group action dynamical systems and show that under the tempered condition, the Bowen entropy of the whole compact space for a given Følner sequence equals to the topological entropy. For the proof of this result, we establish a variational principle related to the Bowen entropy and the Brin-Katok's local entropy formula for dynamical systems with amenable group actions.

math.DS↗
arXiv2016

On large deviations for amenable group actions

By proving an amenable version of Katok's entropy formula and handling the quasi tiling techniques, we establish large deviations bounds for countable discrete amenable group actions. This generalizes the classical results of Lai-Sang Young.

math.DS↗
arXiv2015

Conditional Variational Principle for Historic Set in Some Nonuniformly Hyperbolic Systems

This article is devoted to the study of the historic set, which was introduced by Ruelle, of Birkhoff averges in some nonuniformly hyperbolic systems via Pesin theory. Particularly, we give a conditional variational principle for historic sets. Our results can be applied (i) to the diffeomorphisms on surfaces, (ii) to the nonuniformly hyperbolic diffeomorphisms described by Katok and several other classes of diffeomorphisms derived from Anosov systems.

math.DS↗
arXiv2015

Generic Points in Some Nonuniformly Hyperbolic Systems Via Pesin Theory

This article is devoted to the investigation of the topological pressure of generic points for nonuniformly hyperbolic systems via Pesin theory. In particular, our result can be applied to the nonuniformly hyperbolic diffeomorphisms described by Katok and several other classes of diffeomorphisms derived from Anosov systems.

math.DS↗
arXiv2012

Multifractal analysis for historic set in topological dynamical systems

In this article, the historic set is divided into different level sets and we use topological pressure to describe the size of these level sets. We give an application of these results to dimension theory. Especially, we use topological pressure to describe the relative multifractal spectrum of ergodic averages and give a positive answer to the conjecture posed by L. Olsen (J. Math. Pures Appl. {\bf 82} (2003)).

math.DS↗