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arXiv · 1510.00495

Zero-one law of Hausdorff dimensions of the recurrent sets

Dong Han Kim·Bing Li
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Abstract

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

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Dong Han Kim, Bing Li. 2016-04-04. Zero-one law of Hausdorff dimensions of the recurrent sets. https://arxiv.org/abs/1510.00495

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