Group Extensions for Random Shifts of Finite Type

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 Gurevi\v{c} 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 Gurevi\v{c} entropy of random group $G$ extensions is equal to the relative Gurevi\v{c} 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.