Group Extensions for Random Shifts of Finite Type

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 $G^{\mathrm{ab}}$ by the abelianization of $G$ where $G^{\mathrm{ab}}=G/[G,G]$, we consider the random group extensions of random shifts of finite type between $G$ and $G^{\mathrm{ab}}$. It can be proved that the relative Gurevič entropy of random group $G$ extensions is equal to the relative Gurevič entropy of random group $G^{\mathrm{ab}}$ 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.

42 pages