Measures maximizing the entropy for Kan endomorphisms

TL;DR

This paper extends the understanding of entropy-maximizing measures in Kan endomorphisms, demonstrating the existence of interior hyperbolic measures that maximize entropy, beyond boundary-supported measures.

Contribution

It proves the existence of interior entropy-maximizing measures for a broader class of maps including perturbations of Kan's example, expanding previous results.

Findings

Existence of a third hyperbolic measure in the interior that maximizes entropy.

Extension of entropy-maximizing measures to larger classes of maps and perturbations.

Identification of measures supported in the interior and boundary that maximize entropy.

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 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.