Measures maximizing the entropy for Kan endomorphisms

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 for a larger class of maps including the Kan example. We also prove this statement for a larger class of invariant measures of large class maps including perturbations of the Kan example.