Full flexibility of entropies among ergodic measures for partially hyperbolic diffeomorphisms

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.