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

TL;DR

This paper establishes a sharp upper bound on the number of ergodic measures of maximal entropy for certain partially hyperbolic diffeomorphisms on the 3-torus, based on Morse-Smale dynamics properties.

Contribution

It provides a new upper bound for maximal measures in partially hyperbolic systems with compact center leaves, linking it to Morse-Smale dynamics features.

Findings

Upper bound depends on sources and sinks of Morse-Smale dynamics

Results apply to Kan-type diffeomorphisms with physical measures

Advances understanding of entropy measures in hyperbolic systems

Abstract

In this paper, we study the number of ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms defined on $3-$torus with compact center leaves. Assuming the existence of a periodic leaf with Morse-Smale dynamics we prove a sharp upper bound for the number of maximal measures in terms of the number of sources and sinks of Morse-Smale dynamics. A well-known class of examples for which our results apply are the so-called Kan-type diffeomorphisms admitting physical measures with intermingled basins.