Entropy stability for diffeomorphisms isotopic to Anosov on $\mathbb{T}^d$

Abstract

In this paper we study the behavior of topological entropy for partially hyperbolic diffeomorphisms on $\mathbb{T}^d$ isotopic to a linear Anosov automorphism with indecomposable weak-stable subspace. In particular, we prove that for a class of DA maps associated with such linear systems, whose central behavior is sufficiently dominated by the expansion rate of the linear model, the topological entropy coincides with that of the linear map. Furthermore, we obtain uniqueness of equilibrium states for a class of low-oscillation potentials and, as an application, the uniqueness of the measure of maximal entropy. We also provide sufficient conditions for which ergodic measures arise as the unique equilibrium state of some continuous potential. Finally, on $\mathbb{T}^3$ we construct a dynamically coherent diffeomorphism in the same isotopy class whose topological entropy is strictly larger than that of the linear part.