On the prevalence of the periodicity of maximizing measures
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916198740918272 |
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| author | Ding, Jian Li, Zhiqiang Zhang, Yiwei |
| author_facet | Ding, Jian Li, Zhiqiang Zhang, Yiwei |
| contents | For a continuous map $T: X\rightarrow X$ on a compact metric space $(X,d)$, we say that a function $f: X \rightarrow \mathbb{R}$ has the property $\mathscr{P}_T$ if its time averages along forward orbits of $T$ are maximized at a periodic orbit. In this paper, we prove that for the one-side full shift of two symbols, the property $\mathscr{P}_T$ is prevalent (in the sense of Hunt--Sauer--Yorke) in spaces of Lipschitz functions with respect to metrics with mildly fast decaying rate on the diameters of cylinder sets. This result is a strengthening of \cite[Theorem~A]{BZ16}, confirms the prediction mentioned in the ICM proceeding contribution of J. Bochi (\cite[Seciton 1]{Boc18}) suggested by experimental evidence, and is another step towards the Hunt--Ott conjectures in the area of ergodic optimization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_00536 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the prevalence of the periodicity of maximizing measures Ding, Jian Li, Zhiqiang Zhang, Yiwei Dynamical Systems Probability 37A99 (Primary) 05C80, 37A50, 37C40, 37D35 (Secondary) For a continuous map $T: X\rightarrow X$ on a compact metric space $(X,d)$, we say that a function $f: X \rightarrow \mathbb{R}$ has the property $\mathscr{P}_T$ if its time averages along forward orbits of $T$ are maximized at a periodic orbit. In this paper, we prove that for the one-side full shift of two symbols, the property $\mathscr{P}_T$ is prevalent (in the sense of Hunt--Sauer--Yorke) in spaces of Lipschitz functions with respect to metrics with mildly fast decaying rate on the diameters of cylinder sets. This result is a strengthening of \cite[Theorem~A]{BZ16}, confirms the prediction mentioned in the ICM proceeding contribution of J. Bochi (\cite[Seciton 1]{Boc18}) suggested by experimental evidence, and is another step towards the Hunt--Ott conjectures in the area of ergodic optimization. |
| title | On the prevalence of the periodicity of maximizing measures |
| topic | Dynamical Systems Probability 37A99 (Primary) 05C80, 37A50, 37C40, 37D35 (Secondary) |
| url | https://arxiv.org/abs/2303.00536 |