Taiwanese Journal of Mathematics

Quantitative Recurrence Properties for Systems with Non-uniform Structure

Cao Zhao and Ercai Chen

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Abstract

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.

Article information

Source
Taiwanese J. Math., Volume 22, Number 1 (2018), 225-244.

Dates
Received: 9 March 2017
Revised: 6 April 2017
Accepted: 11 April 2017
First available in Project Euclid: 17 August 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1502935241

Digital Object Identifier
doi:10.11650/tjm/8071

Mathematical Reviews number (MathSciNet)
MR3749362

Zentralblatt MATH identifier
06965367

Subjects
Primary: 37D35: Thermodynamic formalism, variational principles, equilibrium states

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

Citation

Zhao, Cao; Chen, Ercai. Quantitative Recurrence Properties for Systems with Non-uniform Structure. Taiwanese J. Math. 22 (2018), no. 1, 225--244. doi:10.11650/tjm/8071. https://projecteuclid.org/euclid.twjm/1502935241


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