A Conservative Partially Hyperbolic Dichotomy: Hyperbolicity versus Nonhyperbolic Measures

TL;DR

This paper establishes a dichotomy in conservative partially hyperbolic systems: they are either Anosov or have nonhyperbolic ergodic measures, using a perturbation-free approach and new construction techniques.

Contribution

It introduces a perturbation-free method to prove the existence of nonhyperbolic ergodic measures in certain partially hyperbolic systems, expanding understanding of their dynamical behavior.

Findings

Dichotomy between Anosov systems and systems with nonhyperbolic measures

Construction of nonhyperbolic ergodic measures in minimal strong unstable foliation sets

Applicability to non-dynamically coherent subclasses of partially hyperbolic diffeomorphisms

Abstract

In a conservative and partially hyperbolic three-dimensional setting, we study three representative classes of diffeomorphisms: those homotopic to Anosov (or Derived from Anosov diffeomorphisms), diffeomorphisms in neighborhoods of the time-one map of the geodesic flow on a surface of negative curvature, and accessible and dynamically coherent skew products with circle fibers. In any of these classes, we establish the following dichotomy: either the diffeomorphism is Anosov, or it possesses nonhyperbolic ergodic measures. Our approach is perturbation-free and combines recent advances in the study of stably ergodic diffeomorphisms with a variation of the periodic approximation method to obtain ergodic measures. A key result in our construction, independent of conservative hypotheses, is the construction of nonhyperbolic ergodic measures for sets with a minimal strong unstable foliation that satisfy the mostly expanding property. This approach enables us to obtain nonhyperbolic ergodic measures in other contexts, including some subclasses of the so-called anomalous partially hyperbolic diffeomorphisms that are not dynamically coherent.