Group Extensions for Random Shifts of Finite Type

2024/03/20 by Kexiang Yang, Yang, Kexiang, Ercai Chen +5
Mathematics · Computer Science · #Advanced Topology and Set Theory #Mathematical Dynamics and Fractals #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2403.13483

Abstract

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 ℤ^d extensions.

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