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

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.

Equilibrium states for certain partially hyperbolic systems

“…For the setting of partially hyperbolic systems, results in [24] imply that there exists at least one maximal entropy measure if the center bundle is one-dimensional. For C 1`α accessible partially hyperbolic over 3-manifolds having compact one-dimensional central leaves, a dichotomy was proved about the number of maximal entropy measures [50,58]; Rocha and Tahzibi also obtained a similar dichotomy for partially hyperbolic diffeomorphisms defined on 3-torus with compact center leaves [49]. Certain partially hyperbolic systems with one-dimensional center bundles and isotopic to an Anosov diffeomorphism (derived from Anosov (DA)), have a unique maximal entropy measure [13,28,57].…”