Open Access
February, 2018 Quantitative Recurrence Properties for Systems with Non-uniform Structure
Cao Zhao, Ercai Chen
Taiwanese J. Math. 22(1): 225-244 (February, 2018). DOI: 10.11650/tjm/8071


Let $X$ be a subshift with non-uniform structure, and $\sigma \colon X \to X$ be a shift map. Further, define \[ R(\psi) := \{x \in X: d(\sigma^{n}x,x) \lt \psi(n) \textrm{ for infinitely many } n\} \] and \[ R(f) := \left\{ x \in X: d(\sigma^{n}x,x) \lt e^{-S_{n} f(x)} \textrm{ for infinitely many } n \right\}, \] where $\psi \colon \mathbb{N} \to \mathbb{R}^{+}$ is a nonincreasing and positive function and $f \colon X \to \mathbb{R}^{+}$ is a continuous positive function. In this paper, we give quantitative estimates of the above sets, that is, $\dim_{H} R(\psi)$ can be expressed by $\psi$ and $\dim_{H} R(f)$ is the solution of the Bowen equation of topological pressure. These results can be applied to a large class of symbolic systems, including $\beta$-shifts, $S$-gap shifts, and their factors.


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Cao Zhao. Ercai Chen. "Quantitative Recurrence Properties for Systems with Non-uniform Structure." Taiwanese J. Math. 22 (1) 225 - 244, February, 2018.


Received: 9 March 2017; Revised: 6 April 2017; Accepted: 11 April 2017; Published: February, 2018
First available in Project Euclid: 17 August 2017

zbMATH: 06965367
MathSciNet: MR3749362
Digital Object Identifier: 10.11650/tjm/8071

Primary: 37D35

Keywords: Hausdorff dimension , non-uniform structure , recurrence , shrinking target , topology pressure

Rights: Copyright © 2018 The Mathematical Society of the Republic of China

Vol.22 • No. 1 • February, 2018
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