Closest Distance between Iterates of Typical Points
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866916145486888960 |
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| author | Zhao, Boyuan |
| author_facet | Zhao, Boyuan |
| contents | 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 Hölder potential, we prove the asymptotic length of the longest common substring for a typical point converges and the limit depends on the Rényi 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_01628 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Closest Distance between Iterates of Typical Points Zhao, Boyuan Dynamical Systems 37B20, 37B10 (Primary) 60F10 (Secondary) 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 Hölder potential, we prove the asymptotic length of the longest common substring for a typical point converges and the limit depends on the Rényi 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. |
| title | Closest Distance between Iterates of Typical Points |
| topic | Dynamical Systems 37B20, 37B10 (Primary) 60F10 (Secondary) |
| url | https://arxiv.org/abs/2306.01628 |