Equilibrium states for a class of skew products

Abstract

We consider skew products on, where is the two-sphere or the two-torus, which are partially hyperbolic and semi-conjugate to an Axiom A diffeomorphism. This class of dynamics includes the open sets of-non-stable systems introduced by Abraham and Smale [Non-genericity of -stability. Global Analysis (Proceedings of Symposia in Pure Mathematics, XIV (Berkeley 1968)). American Mathematical Society, Providence, RI, 1970, pp. 5-8.] and Shub [Topological Transitive Diffeomorphisms in (Lecture Notes in Mathematics, 206). Springer, Berlin, 1971, pp. 39-40]. We present sufficient conditions, both on the skew products and the potentials, for the existence and uniqueness of equilibrium states, and discuss their statistical stability. (c) Cambridge University Press, 2019.