Ergodic optimization for Gauss's continued fraction map
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911338530340864 |
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| author | Huang, Yinying Jenkinson, Oliver Li, Zhiqiang |
| author_facet | Huang, Yinying Jenkinson, Oliver Li, Zhiqiang |
| contents | The theory of ergodic optimization for distance-expanding maps is extended to Gauss's continued fraction map. Since the set of invariant probability measures is not weak$^*$ closed, we establish a characterisation of the closure of this set, and investigate limit-maximizing measures for Hölder continuous functions. Although a Mañé cohomology lemma is shown to hold, the typical periodic optimization conjecture is shown to fail, as a consequence of the typical finite optimization property established for a certain class of (rationally maximized) functions. The typical periodic optimization (TPO) property is shown to hold, however, for the class of $α$-Hölder essentially compact functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_21394 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ergodic optimization for Gauss's continued fraction map Huang, Yinying Jenkinson, Oliver Li, Zhiqiang Dynamical Systems Primary: 37A99, Secondary: 37A05, 37B25, 37B65, 37C50, 37E05, 37A44 The theory of ergodic optimization for distance-expanding maps is extended to Gauss's continued fraction map. Since the set of invariant probability measures is not weak$^*$ closed, we establish a characterisation of the closure of this set, and investigate limit-maximizing measures for Hölder continuous functions. Although a Mañé cohomology lemma is shown to hold, the typical periodic optimization conjecture is shown to fail, as a consequence of the typical finite optimization property established for a certain class of (rationally maximized) functions. The typical periodic optimization (TPO) property is shown to hold, however, for the class of $α$-Hölder essentially compact functions. |
| title | Ergodic optimization for Gauss's continued fraction map |
| topic | Dynamical Systems Primary: 37A99, Secondary: 37A05, 37B25, 37B65, 37C50, 37E05, 37A44 |
| url | https://arxiv.org/abs/2512.21394 |