Zero-one law of Hausdorff dimensions of the recurrent sets

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 $$E_{\alpha ,\beta }^{\phi }=\left\{x\in \Sigma :\underset{n\to \infty }{\mathrm{liminf}}\frac{\log R_n(x)}{\phi (n)}=\alpha ,\underset{n\to \infty }{\mathrm{limsup}}\frac{\log R_n(x)}{\phi (n)}=\beta \right\},$$ where $\phi :\mathbb{N}\to {\mathbb{R}}^{+}$ is a monotonically increasing function and $0\le \alpha \le \beta \le +\infty$. We show that the Hausdorff dimension of the set $E_{\alpha ,\beta }^{\phi }$ admits a dichotomy: it is either zero or one depending on $\phi ,\alpha$ and $\beta$.

arXiv comment24 pages