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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.19828 |
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| _version_ | 1866916305231151104 |
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| author | Shinoda, Mao Takahasi, Hiroki Yamamoto, Kenichiro |
| author_facet | Shinoda, Mao Takahasi, Hiroki Yamamoto, Kenichiro |
| contents | Ergodic optimization aims to describe dynamically invariant probability measures that maximize the integral of a given function. The Dyck and Motzkin shifts are well-known examples of transitive subshifts over a finite alphabet that are not intrinsically ergodic. We show that the space of continuous functions on any Dyck-Motzkin shift splits into two subsets: one is a dense $G_δ$ set with empty interior for which any maximizing measure has zero entropy; the other is contained in the closure of the set of functions having uncountably many, fully supported measures that are Bernoulli. One key ingredient of a proof of this result is the path connectedness of the space of ergodic measures of the Dyck-Motzkin shift. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19828 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ergodic optimization for continuous functions on the Dyck-Motzkin shifts Shinoda, Mao Takahasi, Hiroki Yamamoto, Kenichiro Dynamical Systems Ergodic optimization aims to describe dynamically invariant probability measures that maximize the integral of a given function. The Dyck and Motzkin shifts are well-known examples of transitive subshifts over a finite alphabet that are not intrinsically ergodic. We show that the space of continuous functions on any Dyck-Motzkin shift splits into two subsets: one is a dense $G_δ$ set with empty interior for which any maximizing measure has zero entropy; the other is contained in the closure of the set of functions having uncountably many, fully supported measures that are Bernoulli. One key ingredient of a proof of this result is the path connectedness of the space of ergodic measures of the Dyck-Motzkin shift. |
| title | Ergodic optimization for continuous functions on the Dyck-Motzkin shifts |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2406.19828 |