Maximal entropy measures of diffeomorphisms of circle fiber bundles

TL;DR

This paper characterizes the maximal entropy measures of certain partially hyperbolic circle bundle diffeomorphisms, revealing a dichotomy in 3D nilmanifolds and showing continuous variation of measures with the dynamics.

Contribution

It provides a novel characterization of maximal entropy measures using skeletons and establishes a dichotomy for 3D nilmanifolds, including uniqueness and hyperbolicity properties.

Findings

Unique maximal measure for rotation extensions of Anosov diffeomorphisms.

Existence of exactly two hyperbolic maximal measures with opposite Lyapunov exponents.

Maximal measures vary continuously with the diffeomorphism.

Abstract

We characterize the maximal entropy measures of partially hyperbolic C^2 diffeomorphisms whose center foliations form circle bundles, by means of suitable finite sets of saddle points, that we call skeletons. In the special case of 3-dimensional nilmanifolds other than the torus, this entails the following dichotomy: either the diffeomorphism is a rotation extension of an Anosov diffeomorphism -- in which case there is a unique maximal measure, with full support and zero center Lyapunov exponents -- or there exist exactly two ergodic maximal measures, both hyperbolic and whose center Lyapunov exponents have opposite signs. Moreover, the set of maximal measures varies continuously with the diffeomorphism.