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Partially hyperbolic diffeomorphisms with one-dimensional neutral center on 3-manifolds

Jinhua Zhang

Abstract

We prove that for any partially hyperbolic diffeomorphism with one dimensional neutral center on a 3-manifold, the center stable and center unstable foliations are complete; moreover, each leaf of center stable and center unstable foliations is a cylinder, a M$\ddot{o}$bius band or a plane. Further properties of the Bonatti-Parwani-Potrie type of partially hyperbolic diffeomorphisms are studied. Such examples are obtained by composing the time $m$-map (for $m>0$ large) of a non-transitive Anosov flow $φ_t$ on an orientable 3-manifold with Dehn twists along some transverse tori, and the examples are partially hyperbolic with one-dimensional neutral center. We prove that the center foliation gives a topologically Anosov flow which is topologically equivalent to $φ_t$. We also prove that for the precise example constructed by Bonatti-Parwani-Potrie, the center stable and center unstable foliations are robustly complete.

Partially hyperbolic diffeomorphisms with one-dimensional neutral center on 3-manifolds

Abstract

We prove that for any partially hyperbolic diffeomorphism with one dimensional neutral center on a 3-manifold, the center stable and center unstable foliations are complete; moreover, each leaf of center stable and center unstable foliations is a cylinder, a Mbius band or a plane. Further properties of the Bonatti-Parwani-Potrie type of partially hyperbolic diffeomorphisms are studied. Such examples are obtained by composing the time -map (for large) of a non-transitive Anosov flow on an orientable 3-manifold with Dehn twists along some transverse tori, and the examples are partially hyperbolic with one-dimensional neutral center. We prove that the center foliation gives a topologically Anosov flow which is topologically equivalent to . We also prove that for the precise example constructed by Bonatti-Parwani-Potrie, the center stable and center unstable foliations are robustly complete.

Paper Structure

This paper contains 15 sections, 27 theorems, 21 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Our setting
  3. Statement of the results
  4. Preliminary
  5. Dynamical coherence, plaque expansiveness and completeness
  6. Existence of compact leaves
  7. Anosov flow
  8. Dehn twists
  9. The topological structure of the center stable and center unstable foliations: Proof of Theorem \ref{['thm.plane and cylinder']}
  10. Center flow carried by the partially hyperbolic diffeomorphisms derived from Dehn surgery: Proof of Theorem \ref{['thm.conjugate to Anosov flow']}
  11. Partially hyperbolic diffeomorphisms from the Dehn surgery of Anosov flows
  12. The anomalous example in BPP: proof of Proposition \ref{['thm.conjugation']}
  13. Construction of the example in BPP
  14. Action of $f_b$ on the space of center leaves intersecting $T_1$
  15. Robust completeness of the center stable foliation of BPP example

Key Result

Theorem A

Let $M$ be a closed 3-manifold and $f\in\mathcal{P}\mathcal{H}(M)$. Assume that $f$ has neutral behavior along the center, then we have the followings:

Figures (5)

  • Figure 1:
  • Figure 2:
  • Figure 3: The real lines and the dash lines denote the leaves of the lifts of foliations $\mathcal{F}^s_i$ and $\mathcal{F}^u_i$ on the universal cover respectively.
  • Figure 4: The real lines and the dash lines denote the leaves of the lifts of foliations $\mathcal{F}^u_i$ and $\mathcal{F}^s_i$ to the universal cover respectively.
  • Figure 5: The dash line denotes the center leaf obtained by the intersection of center stable and center unstable leaves.

Theorems & Definitions (47)

  • Conjecture 1
  • Theorem A
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Theorem B
  • Corollary \oldthetheorem
  • Proposition \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • ...and 37 more