On the continuity of topological entropy of certain partially hyperbolic diffeomorphisms

Abstract

In this paper, we consider certain partially hyperbolic diffeomorphisms (PHDs) with centre of arbitrary dimension and obtain continuity properties of the topological entropy under C 1 perturbations. The systems considered have subexponential growth in the centre direction and uniform exponential growth along the unstable foliation. Our result applies to PHDs which are Lyapunov stable in the centre direction. It applies to another important class of systems which do have subexponential growth in the centre direction, for which we develop a technique to use exponential mixing property of the systems to get uniform distribution of unstable manifolds. A primary example are the translations on homogeneous spaces which may have centre of arbitrary dimension and of polynomial orbit growth.