Shub’s example revisited

Abstract

For a class of robustly transitive diffeomorphisms on ${\mathbb T}^4$ introduced by Shub [Topologically transitive diffeomorphisms of $T^4$ . Proceedings of the Symposium on Differential Equations and Dynamical Systems (Lecture notes in Mathematics, 206). Ed. D. Chillingworth. Springer, Berlin, 1971, pp. 39–40], satisfying an additional bunching condition, we show that there exists a $C^2$ open and $C^r$ dense subset ${\mathcal U}^r$ , $2\leq r\leq \infty $ , such that any two hyperbolic points of $g\in {\mathcal U}^r$ with stable index $2$ are homoclinically related. As a consequence, every $g\in {\mathcal U}^r$ admits a unique homoclinic class associated to the hyperbolic periodic points with index $2$ , and this homoclinic class coincides with the whole ambient manifold. Moreover, every $g\in {\mathcal U}^r$ admits at most one measure of maximal entropy, and every $g\in {\mathcal U}^{\infty }$ admits a unique measure of maximal entropy.