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 $\mu$ be an ergodic $G-$invariant Borel probability measure on $X$. For a fixed tempered F{\o}lner sequence $\{F_n\}$ in $G$ with $\lim\limits_{n\rightarrow+\infty}\frac{|F_n|}{\log n}=\infty$, we prove the following variational principle: $$h^B(G_{\mu},\{F_n\})=h_{\mu}(X,G),$$ where $G_{\mu}$ is the set of generic points for $\mu$ with respect to $\{F_n\}$ and $h^B(G_{\mu},\{F_n\})$ is the Bowen topological entropy (along $\{F_n\}$) on $G_{\mu}$. This generalizes the classical result of Bowen in 1973.Comment: Science China Mathematics, 201

On the topological pressure of the saturated set with non-uniform structure

We derive a conditional variational principle of the saturated set for systems with the non-uniform structure. Our result applies to a broad class of systems including beta-shifts, S-gap shifts and their factors.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1605.07283; text overlap with arXiv:1304.5497 by other author

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.Comment: 22pages. arXiv admin note: substantial text overlap with arXiv:1502.02459, arXiv:1412.076

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)).Comment: 30 page

Projection Pressure and Bowen's Equation for a Class of Self-similar Fractals with Overlap Structure

Let $\{S_i\}_{i=1}^{l}$ be an iterated function system(IFS) on $\mathbb{R}^d$ with attractor K. Let $\pi$ be the canonical projection. In this paper we define a new concept called "projection pressure" $P_\pi(\phi)$ for $\phi\in C(\mathbb{R}^d)$ under certain affine IFS, and show the variational principle about the projection pressure. Furthermore we check that the unique zero root of "projection pressure" still satisfies Bowen's equation when each $S_i$ is the similar map with the same compression ratio. Using the root of Bowen's equation, we can get the Hausdorff dimension of the attractor $K$

The variational principle of local pressure for actions of sofic group

This study establishes the variational principle for local pressure in the sofic context.Comment: 13 page

Relative tail entropy for random bundle transformations

We introduce the relative tail entropy to establish a variational principle for continuous bundle random dynamical systems. We also show that the relative tail entropy is conserved by the principal extension

Induced topological pressure for topological dynamical(to appear in JPM)

In this paper, inspired by the article [5], we introduce the induced topological pressure for a topological dynamical system. In particular, we prove a variational principle for the induced topological pressure

The Bowen's topological entropy of the Cartesian product sets

This article is devoted to showing the product theorem for Bowen's topological entropy.Comment: 11 pages. arXiv admin note: text overlap with arXiv:1012.1103 by other author

Packing dimensions of the divergence points of self-similar measures with the open set condition

Let $\mu$ be the self-similar measure supported on the self-similar set $K$ with open set condition. In this article, we discuss the packing dimension of the set $\{x\in K: A(\frac{\log\mu(B(x,r))}{\log r})=I\}$ for $I\subseteq\mathbb{R}$, where $A(\frac{\log\mu(B(x,r))}{\log r})$ denotes the set of accumulation points of \frac{\log\mu(B(x,r))}{\log r}$as$r\searrow0$. Our main result solves the conjecture about packing dimension posed by Olsen and Winter \cite{OlsWin} and generalizes the result in \cite{BaeOlsSni}.Comment: 13 page