邀请人：陈二才教授，周效尧副教授
摘要：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 Gureviˇc pressure(entropy), the spectral radius of random Perron-Frobenius operator and amenability of G. Under the assumption of the topologically mixing and relative BIP property, we prove that the relative Gureviˇc pressure of random group extensions is equal to the relative Gureviˇc pressure of random countable Markov shifts implies that G is amenable. Moreover, the amenability of G can be characterized by the spectral radius of random Perron-Frobenius operator of random group extensions, we prove that the logarithm of the spectral radius of this random Perron-Frobenius operator coincides with the relative Gureviˇc pressure of random countable Markov shifts almost everywhere if and only if G is amenable. Given Gab by the abelianization of G where Gab = G/[G, G], we consider the random group extensions between G and Gab. If G is amenable, we prove that the relative Gureviˇc pressure of random group G extensions is given by the relative Gureviˇc pressure of random group Gab extensions for random shifts of finite type. In addition, we prove that the relative Gureviˇc entropy of random group G extensions is equal to the relative Gureviˇc entropy of random group Gab extensions if and only if G is amenable.
