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arXiv · 2306.01628

Closest Distance between Iterates of Typical Points

Boyuan Zhao
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Abstract

The shortest distance between the first $n$ iterates of a typical point can be quantified with a log rule for some dynamical systems admitting Gibbs measures. We show this in two settings. For topologically mixing Markov shifts with at most countably infinite alphabet admitting a Gibbs measure with respect to a locally H\"{o}lder potential, we prove the asymptotic length of the longest common substring for a typical point converges and the limit depends on the R\'{e}nyi entropy. For interval maps with the Gibbs-Markov structure, we prove a similar rule relating the correlation dimension of Gibbs measures with the shortest distance between two iterates in the orbit generated by a typical point.

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Boyuan Zhao. 2023-06-02. Closest Distance between Iterates of Typical Points. https://arxiv.org/abs/2306.01628

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