Zero-one law of Hausdorff dimensions of the recurrent sets

Dong Han  Kim; Bing  Li

doi:10.3934/dcds.2016041

Article Contents

2016, Volume 36, Issue 10: 5477-5492. Doi: 10.3934/dcds.2016041

This issue Previous Article On one dimensional quantum Zakharov system Next Article Well-posedness and blow-up phenomena for a generalized Camassa-Holm equation

Zero-one law of Hausdorff dimensions of the recurrent sets

Dong Han Kim^1, and
Bing Li^2,

1.
Department of Mathematics Education, Dongguk University - Seoul, Seoul 04620
2.
Department of Mathematics, South China University of Technology, Guangzhou, 510641

Received: September 30, 2015

Revised: February 29, 2016

Published: May 2017

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Abstract

Let $(\Sigma, \sigma)$ be the one-sided shift space with $m$ symbols and $R_n(x)$ be the first return time of $x\in\Sigma$ to the $n$-th cylinder containing $x$. Denote $$E^\varphi_{\alpha,\beta}=\left\{x\in\Sigma: \liminf_{n\to\infty}\frac{\log R_n(x)}{\varphi(n)}=\alpha,\ \limsup_{n\to\infty}\frac{\log R_n(x)}{\varphi(n)}=\beta\right\},$$ where $\varphi: \mathbb{N}\to \mathbb{R}^+$ is a monotonically increasing function and $0\leq\alpha\leq\beta\leq +\infty$. We show that the Hausdorff dimension of the set $E^\varphi_{\alpha,\beta}$ admits a dichotomy: it is either zero or one depending on $\varphi, \alpha$ and $\beta$.

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Mathematics Subject Classification: 28A80.

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