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 Gurevic pressure (entropy), the spectral radius of random Perron-Frobenius operator and amenability of G. Given G^ ab by the abelianization of G where G^ ab=G/G, G, we consider the random group extensions of random shifts of finite type between G and G^ ab. It can be proved that the relative Gurevic entropy of random group G extensions is equal to the relative Gurevic entropy of random group G^ 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.