Non-compact spaces of invariant measures
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909536217989120 |
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| author | Iommi, Godofredo Velozo, Anibal |
| author_facet | Iommi, Godofredo Velozo, Anibal |
| contents | We study a compactification of the space of invariant probability measures for a transitive countable Markov shift. We prove that it is affine homeomorphic to the Poulsen simplex. Furthermore, we establish that, depending on a combinatorial property of the shift space, the compactification contains either a single new ergodic measure or a dense set of them. As an application of our results, we prove that the space of ergodic probability measures of a transitive countable Markov shift is homeomorphic to $\ell_2$, extending to the non-compact setting a known result for subshifts of finite type. Additionally, we explore implications for thermodynamic formalism, including a version of the dual variational principle for transitive countable Markov shifts with uniformly continuous potentials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09836 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-compact spaces of invariant measures Iommi, Godofredo Velozo, Anibal Dynamical Systems We study a compactification of the space of invariant probability measures for a transitive countable Markov shift. We prove that it is affine homeomorphic to the Poulsen simplex. Furthermore, we establish that, depending on a combinatorial property of the shift space, the compactification contains either a single new ergodic measure or a dense set of them. As an application of our results, we prove that the space of ergodic probability measures of a transitive countable Markov shift is homeomorphic to $\ell_2$, extending to the non-compact setting a known result for subshifts of finite type. Additionally, we explore implications for thermodynamic formalism, including a version of the dual variational principle for transitive countable Markov shifts with uniformly continuous potentials. |
| title | Non-compact spaces of invariant measures |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2503.09836 |