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A Conservative Partially Hyperbolic Dichotomy: Hyperbolicity versus Nonhyperbolic Measures

Lorenzo J. Díaz, Jiagang Yang, Jinhua Zhang

TL;DR

The paper analyzes conservative, partially hyperbolic diffeomorphisms in dimension three with a one-dimensional center, proving a dichotomy: either the dynamics are uniformly hyperbolic (Anosov) or the system supports nonhyperbolic ergodic measures. It develops a perturbation-free framework that combines recent stability ergodicity results with a generalized GIKN periodic-approximation method, aided by Liao shadowing and $u$-Gibbs state theory under the mostly expanding condition. A central technical contribution is a general criterion: if a set is $\,\mathcal{F}^u$-minimal, mostly expanding, saturated, and contains a negative center-exponent periodic point, then a nonhyperbolic ergodic measure supported on that set exists. The results cover three representative classes (DA systems on $\,\mathbb{T}^3$, time-one maps of geodesic flows, and accessible skew products with circle fibers), with broader implications via a transversality-based criterion that can apply beyond dynamical coherence. Collectively, the work strengthens Palis-type dichotomies in conservative partially hyperbolic dynamics and illuminates the mechanisms by which nonhyperbolicity emerges in structured 3D systems, including anomalous examples.

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.

A Conservative Partially Hyperbolic Dichotomy: Hyperbolicity versus Nonhyperbolic Measures

TL;DR

The paper analyzes conservative, partially hyperbolic diffeomorphisms in dimension three with a one-dimensional center, proving a dichotomy: either the dynamics are uniformly hyperbolic (Anosov) or the system supports nonhyperbolic ergodic measures. It develops a perturbation-free framework that combines recent stability ergodicity results with a generalized GIKN periodic-approximation method, aided by Liao shadowing and -Gibbs state theory under the mostly expanding condition. A central technical contribution is a general criterion: if a set is -minimal, mostly expanding, saturated, and contains a negative center-exponent periodic point, then a nonhyperbolic ergodic measure supported on that set exists. The results cover three representative classes (DA systems on , time-one maps of geodesic flows, and accessible skew products with circle fibers), with broader implications via a transversality-based criterion that can apply beyond dynamical coherence. Collectively, the work strengthens Palis-type dichotomies in conservative partially hyperbolic dynamics and illuminates the mechanisms by which nonhyperbolicity emerges in structured 3D systems, including anomalous examples.

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.

Paper Structure

This paper contains 28 sections, 30 theorems, 69 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A question and our results
  3. Context
  4. Approach and ingredients
  5. Statements of the main results
  6. Preliminaries
  7. Main results
  8. A general criterion for nonhyperbolic measures
  9. Construction of nonhyperbolic measures: Proof of Theorem \ref{['thm.existence']}
  10. The GIKN criterion
  11. Liao's shadowing lemma
  12. $u$-Gibbs states and mostly expanding diffeomorphisms
  13. Good periodic approximations
  14. Proof of Proposition \ref{['main.prop']}
  15. End of the Proof of Theorem \ref{['thm.existence']}
  16. ...and 13 more sections

Key Result

Theorem 1

For every $f\in\operatorname{PH}_{ \mathrm{Leb}, c=1}^{1+\alpha}(\mathbb{T}^3)$ homotopic to a linear Anosov diffeomorphism the following dichotomy holds:

Figures (2)

  • Figure 1: Red lines denote strong stable plaques, black lines denote the strong unstable plaques and the green lines denote the center plaque.
  • Figure 2: Blue lines represents the intersection of $\mathcal{F}^{cu}(x_0)$ with $\mathcal{F}^s(\Lambda).$

Theorems & Definitions (55)

  • Remark 2.1
  • Theorem 1
  • Theorem 2
  • Remark 2.2
  • Theorem 3
  • Corollary 4
  • Theorem 5
  • Corollary 6
  • Remark 2.3: Anomalous partially hyperbolic diffeomorphisms
  • Theorem 7
  • ...and 45 more