Group Extensions for Random Shifts of Finite Type
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929282916286464 |
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| author | Yang, Kexiang Chen, Ercai Lin, Zijie Zhou, Xiaoyao |
| author_facet | Yang, Kexiang Chen, Ercai Lin, Zijie Zhou, Xiaoyao |
| contents | Symbolic dynamical theory plays an important role in the research of amenability with a countable group. Motivated by the deep results of Dougall and Sharp, we study the group extensions for topologically mixing random shifts of finite type. For a countable group $G$, we consider the potential connections between relative Gurevič pressure (entropy), the spectral radius of random Perron-Frobenius operator and amenability of $G$. Given $G^{\rm ab}$ by the abelianization of $G$ where $G^{\rm ab}=G/[G,G]$, we consider the random group extensions of random shifts of finite type between $G$ and $G^{\rm ab}$. It can be proved that the relative Gurevič entropy of random group $G$ extensions is equal to the relative Gurevič entropy of random group $G^{\rm ab}$ extensions if and only if $G$ is amenable. Moreover, we establish the relativized variational principle and discuss the unique equilibrium state for random group $\mathbb{Z}^{d}$ extensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_13483 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Group Extensions for Random Shifts of Finite Type Yang, Kexiang Chen, Ercai Lin, Zijie Zhou, Xiaoyao Dynamical Systems Symbolic dynamical theory plays an important role in the research of amenability with a countable group. Motivated by the deep results of Dougall and Sharp, we study the group extensions for topologically mixing random shifts of finite type. For a countable group $G$, we consider the potential connections between relative Gurevič pressure (entropy), the spectral radius of random Perron-Frobenius operator and amenability of $G$. Given $G^{\rm ab}$ by the abelianization of $G$ where $G^{\rm ab}=G/[G,G]$, we consider the random group extensions of random shifts of finite type between $G$ and $G^{\rm ab}$. It can be proved that the relative Gurevič entropy of random group $G$ extensions is equal to the relative Gurevič entropy of random group $G^{\rm ab}$ extensions if and only if $G$ is amenable. Moreover, we establish the relativized variational principle and discuss the unique equilibrium state for random group $\mathbb{Z}^{d}$ extensions. |
| title | Group Extensions for Random Shifts of Finite Type |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2403.13483 |