Ergodic optimization for continuous functions on non-Markov shifts
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| Format: | Preprint |
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2024
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| _version_ | 1866913030363676672 |
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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. For a wide class of intrinsically ergodic subshifts over a finite alphabet, we show that the space of continuous functions on the shift space splits into two subsets: one is a $G_δ$ dense set for which all maximizing measures have `relatively small' entropy; the other is contained in the closure of the set of functions having uncountably many, fully supported ergodic measures with `relatively large' entropy. This result considerably generalizes and unifies the results of Morris (2010) and Shinoda (2018), and applies to a wide class of intrinsically ergodic non-Markov symbolic dynamics without Bowen's specification property, including any transitive piecewise monotonic interval map, some coded shifts and multidimensional $β$-transformations. Along with these examples of application, we provide an example of an intrinsically ergodic subshift with positive obstruction entropy to specification. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_01123 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ergodic optimization for continuous functions on non-Markov shifts Shinoda, Mao Takahasi, Hiroki Yamamoto, Kenichiro Dynamical Systems 37B10, 37D35 Ergodic optimization aims to describe dynamically invariant probability measures that maximize the integral of a given function. For a wide class of intrinsically ergodic subshifts over a finite alphabet, we show that the space of continuous functions on the shift space splits into two subsets: one is a $G_δ$ dense set for which all maximizing measures have `relatively small' entropy; the other is contained in the closure of the set of functions having uncountably many, fully supported ergodic measures with `relatively large' entropy. This result considerably generalizes and unifies the results of Morris (2010) and Shinoda (2018), and applies to a wide class of intrinsically ergodic non-Markov symbolic dynamics without Bowen's specification property, including any transitive piecewise monotonic interval map, some coded shifts and multidimensional $β$-transformations. Along with these examples of application, we provide an example of an intrinsically ergodic subshift with positive obstruction entropy to specification. |
| title | Ergodic optimization for continuous functions on non-Markov shifts |
| topic | Dynamical Systems 37B10, 37D35 |
| url | https://arxiv.org/abs/2406.01123 |