Extensibility and denseness of periodic semigroup actions
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arXiv
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| Format: | Preprint |
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2025
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| author | Briceño, Raimundo Bustos-Gajardo, Álvaro Donoso-Echenique, Miguel |
| author_facet | Briceño, Raimundo Bustos-Gajardo, Álvaro Donoso-Echenique, Miguel |
| contents | We study periodic points and finitely supported invariant measures for continuous semigroup actions. Introducing suitable notions of periodicity in both topological and measure-theoretical contexts, we analyze the space of invariant Borel probability measures associated with these actions. For embeddable semigroups, we establish a direct relationship between the extensibility of invariant measures to the free group on the semigroup and the denseness of finitely supported invariant measures. Applying this framework to shift actions on the full shift, we prove that finitely supported invariant measures are dense for every left amenable semigroup that is residually a finite group and for every finite-rank free semigroup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_00312 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extensibility and denseness of periodic semigroup actions Briceño, Raimundo Bustos-Gajardo, Álvaro Donoso-Echenique, Miguel Dynamical Systems Group Theory Probability Primary 22D40, 37A15, 20M05, Secondary 20M30, 37B10, 60J10 We study periodic points and finitely supported invariant measures for continuous semigroup actions. Introducing suitable notions of periodicity in both topological and measure-theoretical contexts, we analyze the space of invariant Borel probability measures associated with these actions. For embeddable semigroups, we establish a direct relationship between the extensibility of invariant measures to the free group on the semigroup and the denseness of finitely supported invariant measures. Applying this framework to shift actions on the full shift, we prove that finitely supported invariant measures are dense for every left amenable semigroup that is residually a finite group and for every finite-rank free semigroup. |
| title | Extensibility and denseness of periodic semigroup actions |
| topic | Dynamical Systems Group Theory Probability Primary 22D40, 37A15, 20M05, Secondary 20M30, 37B10, 60J10 |
| url | https://arxiv.org/abs/2502.00312 |