Measures maximizing the entropy for Kan endomorphisms

Abstract

In 1994, Ittai Kan provided the first examples of maps with intermingled basins. The Kan example corresponds to a partially hyperbolic endomorphism defined on a surface with the boundary exhibiting two intermingled hyperbolic physical measures. Both measures are supported on the boundary, and they are also measures maximizing the topological entropy. In this work, we prove the existence of a third hyperbolic measure supported in the interior of the cylinder that maximizes the entropy. We prove this statement for a larger class of maps including the Kan example.

On the number of ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms with compact center leaves

“…We also give a topologically transitive example with three ergodic measures of maximal entropy. The construction is based on examples of non-invertible maps appeared in the work of Nuñez-Madariaga, S. Ramirez and C. Vasquez [19]. The celebrated Kan example in the annulus is an endomorphism defined on S 1 × [0, 1] with two physical measures with intermingled basins.…”