Zero-one law of Hausdorff dimensions of the recurrent sets

TL;DR

This paper establishes a zero-one law for the Hausdorff dimensions of certain recurrent sets in symbolic dynamics, showing the dimension is either zero or one depending on specific parameters.

Contribution

It proves a dichotomy for the Hausdorff dimension of sets defined by recurrence rates in shift spaces, extending understanding of dimension theory in dynamical systems.

Findings

Hausdorff dimension of sets $E^ ext{varphi}_{ ext{alpha, beta}}$ is either zero or one.

Dimension depends on the parameters $ ext{varphi}, ext{alpha}, ext{beta}$ in a sharp dichotomy.

Provides a criterion for the dimension based on recurrence behavior in symbolic dynamics.

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 $\beta$.