Cyclicity, hypercyclicity and randomness in self-similar groups
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915758194294784 |
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| author | Fariña-Asategui, Jorge |
| author_facet | Fariña-Asategui, Jorge |
| contents | We introduce the concept of cyclicity and hypercyclicity in self-similar groups as an analogue of cyclic and hypercyclic vectors for an operator on a Banach space. We derive a sufficient condition for cyclicity of non-finitary automorphisms in contracting discrete automata groups. In the profinite setting we prove that fractal profinite groups may be regarded as measure-preserving dynamical systems and derive a sufficient condition for the ergodicity and the mixing properties of these dynamical systems. Furthermore, we show that a Haar-random element in a super strongly fractal profinite group is hypercyclic almost surely as an application of Birkhoff's ergodic theorem for free semigroup actions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11806 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cyclicity, hypercyclicity and randomness in self-similar groups Fariña-Asategui, Jorge Group Theory Dynamical Systems Primary: 20E08, 37A30. Secondary: 20P05, 22F50 We introduce the concept of cyclicity and hypercyclicity in self-similar groups as an analogue of cyclic and hypercyclic vectors for an operator on a Banach space. We derive a sufficient condition for cyclicity of non-finitary automorphisms in contracting discrete automata groups. In the profinite setting we prove that fractal profinite groups may be regarded as measure-preserving dynamical systems and derive a sufficient condition for the ergodicity and the mixing properties of these dynamical systems. Furthermore, we show that a Haar-random element in a super strongly fractal profinite group is hypercyclic almost surely as an application of Birkhoff's ergodic theorem for free semigroup actions. |
| title | Cyclicity, hypercyclicity and randomness in self-similar groups |
| topic | Group Theory Dynamical Systems Primary: 20E08, 37A30. Secondary: 20P05, 22F50 |
| url | https://arxiv.org/abs/2411.11806 |