Shift endomorphisms and compact Lie extensions

Abstract

We consider skew-products with an arbitrary compact Lie group, when the base map is a one-sided shift of finite type endowed with an equilibrium state of a Holder continuous function. First we show that the weak-mixing property of the skew-product implies exactness and exponential mixing. Then we address the problem of classification under measure-theoretic isomorphisms. We show that for a generic set of equilibrium states the isomorphism class of the skew-products corresponds essentially to the cohomology classes of the defining skewing function and the isomorphism is essentially a homeomorphism.