Shub’s example revisited

For a class of robustly transitive diffeomorphisms on T⁴ introduced by Shub [Topologically transitive diffeomorphisms of T⁴ . 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 C² open and Cʳ dense subset Uʳ , 2≤ r≤ ∞ , such that any two hyperbolic points of g∈ Uʳ with stable index $2$ are homoclinically related. As a consequence, every g∈ Uʳ 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∈ Uʳ admits at most one measure of maximal entropy, and every g∈ U∞ admits a unique measure of maximal entropy.