2025/01/09 by Hanfeng Li, Klaus Schmidt, Li, Hanfeng +1
Mathematics · #37A15 (Primary) 37A20 #37A35 #37B10 #37B51 (Secondary) #Advanced Algebra and Geometry #Dynamical Systems (math.DS) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2501.05261
openalex publication_date 2025/01/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Γ be an infinite discrete group and A⊂ Γ a nonempty finite subset. The set of permutations σ of Γ such that s-1σ(s)∈ A for every s∈ Γ can be identified with a shift of finite type XA⊂ AΓ 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 \textrmh_\textrmtop(XA) 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 \textrmper(f) for nonnegative elements f in the real group ring ℝΓ of Γ whose support is the alphabet A of the shift space XA, and compare, for arbitrary f ∈ ℝΓ, the Fuglede-Kadison determinant \textrmdet _\textrmFK(f) with the permanent \textrmper(|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 \textrmper(f) of a nonnegative element f∈ ℝΓ can be viewed as topological pressure of the restricted-permutation shift space XA associated with the function log f on the alphabet A=\textrmsupp(f) of XA.