Restricted Permutations and Permanents of Infinite Amenable Groups
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| Formato: | Preprint |
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2025
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| _version_ | 1866912182107635712 |
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| author | Li, Hanfeng Schmidt, Klaus |
| author_facet | Li, Hanfeng Schmidt, Klaus |
| contents | Let $Γ$ be an infinite discrete group and $\mathsf{A}\subset Γ$ a nonempty finite subset. The set of permutations $σ$ of $Γ$ such that $s^{-1}σ(s)\in \mathsf{A}$ for every $s\in Γ$ can be identified with a shift of finite type $X_\mathsf{A}\subset \mathsf{A}^Γ$ over $Γ$. In this paper we study dynamical properties of such shift spaces, like invariant probability measures, topological entropy, and topological pressure, under the hypothesis that $Γ$ is amenable.
In this case the topological entropy $\textrm{h}_{\textrm{top}}(X_\mathsf{A})$ can be expressed as logarithmic growth rate of permanents of certain finite (0,1)-matrices associated with right Følner sequences in $Γ$. Motivated by the difficulty of computing such permanents we introduce the notion of the permanent $\textrm{per}(f)$ for nonnegative elements $f$ in the real group ring $\mathbb{R}Γ$ of $Γ$ whose support is the alphabet $\mathsf{A}$ of the shift space $X_\mathsf{A}$, and compare, for arbitrary $f \in \mathbb{R}Γ$, the Fuglede-Kadison determinant $\textrm{det} _\textrm{FK}(f)$ with the permanent $\textrm{per}(|f|)$ of the absolute value $|f|$ of $f$. Although this approach is effective in only few examples, discussed below, it is interesting from a conceptual point of view that the permanent $\textrm{per}(f)$ of a nonnegative element $f\in \mathbb{R}Γ$ can be viewed as topological pressure of the restricted-permutation shift space $X_\mathsf{A}$ associated with the function $\log f$ on the alphabet $\mathsf{A}=\textrm{supp}(f)$ of $X_\mathsf{A}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_05261 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Restricted Permutations and Permanents of Infinite Amenable Groups Li, Hanfeng Schmidt, Klaus Dynamical Systems 37A15 (Primary) 37A20, 37A35, 37B10, 37B51 (Secondary) Let $Γ$ be an infinite discrete group and $\mathsf{A}\subset Γ$ a nonempty finite subset. The set of permutations $σ$ of $Γ$ such that $s^{-1}σ(s)\in \mathsf{A}$ for every $s\in Γ$ can be identified with a shift of finite type $X_\mathsf{A}\subset \mathsf{A}^Γ$ over $Γ$. In this paper we study dynamical properties of such shift spaces, like invariant probability measures, topological entropy, and topological pressure, under the hypothesis that $Γ$ is amenable. In this case the topological entropy $\textrm{h}_{\textrm{top}}(X_\mathsf{A})$ can be expressed as logarithmic growth rate of permanents of certain finite (0,1)-matrices associated with right Følner sequences in $Γ$. Motivated by the difficulty of computing such permanents we introduce the notion of the permanent $\textrm{per}(f)$ for nonnegative elements $f$ in the real group ring $\mathbb{R}Γ$ of $Γ$ whose support is the alphabet $\mathsf{A}$ of the shift space $X_\mathsf{A}$, and compare, for arbitrary $f \in \mathbb{R}Γ$, the Fuglede-Kadison determinant $\textrm{det} _\textrm{FK}(f)$ with the permanent $\textrm{per}(|f|)$ of the absolute value $|f|$ of $f$. Although this approach is effective in only few examples, discussed below, it is interesting from a conceptual point of view that the permanent $\textrm{per}(f)$ of a nonnegative element $f\in \mathbb{R}Γ$ can be viewed as topological pressure of the restricted-permutation shift space $X_\mathsf{A}$ associated with the function $\log f$ on the alphabet $\mathsf{A}=\textrm{supp}(f)$ of $X_\mathsf{A}$. |
| title | Restricted Permutations and Permanents of Infinite Amenable Groups |
| topic | Dynamical Systems 37A15 (Primary) 37A20, 37A35, 37B10, 37B51 (Secondary) |
| url | https://arxiv.org/abs/2501.05261 |