Masato Tsujii: Virtually expanding dynamics - a generalization of expanding maps from the view point of micro-local analysis

Abstract: We consider a class of dynamical system "virtually expanding maps". This is a generalization of the class of (uniformly) expanding maps from the view point of micro-local analysis. It forms an open set of maps that contains some partially hyperbolic maps as well as expanding maps. We show that the essential spectral radius of Perron-Frobenius operators for virtually expanding dynamical system is smaller than 1 (as in the case of expanding maps), when they act on a Sobolev space. In particular, each virtually expanding map admits finitely many absolutely continuous SRB measures and we can observe (relatively mild) bifurcations of them inside the class of virtually expanding maps.