Full flexibility of entropies among ergodic measures for partially hyperbolic diffeomorphisms

TL;DR

This paper demonstrates that for a broad class of partially hyperbolic diffeomorphisms with a one-dimensional center, the entropy and center Lyapunov exponent can vary freely within certain bounds, showing high flexibility in their ergodic measures.

Contribution

It establishes the joint flexibility of entropy and center Lyapunov exponent for many nonhyperbolic, transitive partially hyperbolic systems with a one-dimensional center.

Findings

Flexibility of entropy and Lyapunov exponent in these systems.

Applicability to various examples including fibered, flow-type, and derived from Anosov diffeomorphisms.

Formalization of the interplay between center expanding and contracting regions.

Abstract

We study nonhyperbolic and transitive partially hyperbolic diffeomorphisms having a one-dimensional center. We prove joint flexibility with respect to entropy and center Lyapunov exponent for a broad class of these systems. Flexibility means that for any given value of the center Lyapunov exponent and any value of entropy less than the supremum of entropies of ergodic measures with that exponent, there is an ergodic measure with exactly this entropy and exponent. Our hypotheses involve minimal foliations and blender-horseshoes, they formalize the interplay between two regions of the ambient space, one of center expanding and the other of center contracting type. The list of examples our results apply is rather long, a non-exhaustive list includes fibered by circles, flow-type, some Derived from Anosov diffeomorphisms, and some anomalous (non-dynamically coherent) diffeomorphisms.