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Partial hyperbolicity and pseudo-Anosov dynamics

Sergio R. Fenley, Rafael Potrie

Abstract

We show that if a hyperbolic 3-manifold admits a partially hyperbolic diffeomorphism then it also admits an Anosov flow. Moreover, we give a complete classification of partially hyperbolic diffeomorphism in hyperbolic 3-manifolds as well as partially hyperbolic diffeomorphisms in Seifert manifolds inducing pseudo-Anosov dynamics in the base. This classification is given in terms of the structure of their center (branching) foliations and the notion of collapsed Anosov flows.

Partial hyperbolicity and pseudo-Anosov dynamics

Abstract

We show that if a hyperbolic 3-manifold admits a partially hyperbolic diffeomorphism then it also admits an Anosov flow. Moreover, we give a complete classification of partially hyperbolic diffeomorphism in hyperbolic 3-manifolds as well as partially hyperbolic diffeomorphisms in Seifert manifolds inducing pseudo-Anosov dynamics in the base. This classification is given in terms of the structure of their center (branching) foliations and the notion of collapsed Anosov flows.

Paper Structure

This paper contains 52 sections, 60 theorems, 24 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Collapsed Anosov flows
  3. Statements
  4. Tools developed in this article
  5. Idea of proof
  6. More general results
  7. Context and comments
  8. Organization of the paper
  9. Intersection and dependence on previous works
  10. Preliminaries and discussions on some notions
  11. Branching foliations
  12. $\hbox{$\mathbb{R}$}$-covered foliations and hyperbolic metrics
  13. Boundary at infinity and visual metric
  14. The universal circle
  15. Visual metrics on the universal circle $S^1_{univ}$
  16. ...and 37 more sections

Key Result

Theorem A

Let $M$ be a closed hyperbolic 3-manifold admitting a partially hyperbolic diffeomorphism. Then, $M$ admits an Anosov flow.

Figures (9)

  • Figure 1: A pseudo-Anosov pair with $p = 3$.
  • Figure 2: The core of the pA pair.
  • Figure 3: Proof of landing. The iterates $P^n$ push the leaves away from the middle point and into a compact part (when projected to $N$).
  • Figure 4: A configuration.
  • Figure 5: The shadow.
  • ...and 4 more figures

Theorems & Definitions (163)

  • Theorem A
  • Definition 1.1: Collapsed Anosov flow
  • Theorem B
  • Theorem C
  • Remark 1.2
  • Theorem D
  • Theorem E
  • Theorem 2.1: BI
  • Remark 2.2
  • Proposition 2.3
  • ...and 153 more