  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.
