The amount of nonhyperbolicity for partially hyperbolic diffeomorphisms

TL;DR

This paper investigates the nonhyperbolic behavior of partially hyperbolic diffeomorphisms with a one-dimensional center, analyzing entropy and Lyapunov exponents, and constructing specific sets to understand zero-exponent cases.

Contribution

It introduces a framework to measure nonhyperbolicity via entropy and Lyapunov exponents without requiring dynamical coherence, including new constructions for zero-exponent level sets.

Findings

Entropy varies continuously with the Lyapunov exponent.

Maximal entropy is achieved on a set foliated by central curves.

Finite-time Lyapunov exponents converge uniformly to zero.

Abstract

We study the amount of nonhyperbolicity within a broad class of (nonhyperbolic) partially hyperbolic diffeomorphisms with a one-dimensional center. For that, we focus on the center Lyapunov exponent and the entropy of its level sets. We show that these entropies vary continuously and can be expressed in terms of restricted variational principles. In this study, no dynamical coherence is required. Of particular interest is the case where the exponent is zero. To study this level set, we construct a compact set foliated by curves tangent to the central direction. Within this set, the entropy attains the maximal possible (and positive) value. Moreover, finite-time Lyapunov exponents converge uniformly to zero. In this construction, we introduce a mechanism to concatenate center curves. The class studied consists of those robustly transitive diffeomorphisms that have a pair of blender-horseshoes with different types of hyperbolicity and possess minimal strong stable and unstable foliations. This classes includes flow-type and circle-fibered diffeomorphisms as well as some derived from Anosov diffeomorphisms. It also includes the so-called anomalous examples which are dynamically incoherent.