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Partially Hyperbolic Dynamics on $\mathbb T^4$: Existence of Compact Center-Unstable Leaves

Raul Ures, Tongyao Yu

Abstract

We show that for $n\ge2$, if a partially hyperbolic diffeomorphism $f:\mathbb T^{n+1}\to \mathbb T^{n+1}$ with $\dim E^s=\dim E^c=1$ has an invariant center-unstable foliation with a compact incompressible leaf, then this foliation has a transverse closed curve in the universal cover. Also, if $f$ is leaf conjugate to its linear part, it has no compact incompressible center-unstable submanifold. In particular, by the incompressibility result we obtained on Anosov tori, the incompressibility assumptions can be removed when $f$ is defined on $\mathbb T^4$.

Partially Hyperbolic Dynamics on $\mathbb T^4$: Existence of Compact Center-Unstable Leaves

Abstract

We show that for , if a partially hyperbolic diffeomorphism with has an invariant center-unstable foliation with a compact incompressible leaf, then this foliation has a transverse closed curve in the universal cover. Also, if is leaf conjugate to its linear part, it has no compact incompressible center-unstable submanifold. In particular, by the incompressibility result we obtained on Anosov tori, the incompressibility assumptions can be removed when is defined on .
Paper Structure (12 sections, 24 theorems, 28 equations)

This paper contains 12 sections, 24 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Statement of Results
  3. In Lower Dimensions
  4. In Higher Dimensions
  5. Organization of the Paper
  6. Codimension One Partial Hyperbolicity
  7. Compact Center-Unstable Submanifolds
  8. Restrictions on the underlying manifolds
  9. Periodicity, Hyperbolicity, and Incompressibility of Compact Center-unstable Submanifolds
  10. Existence of Compact Leaves
  11. Proof of Theorem \ref{['MainTheorem']}
  12. On Incompressibility of Anosov Tori

Key Result

Theorem 1.3

If there is an $f$-invariant $cu$-foliation $\mathcal{W}^{cu}$ having a compact leaf $T$, there exists a closed curve transverse to the lift $\tilde{\mathcal{W}}^{cu}$. Hence if the leaves of $\tilde{\mathcal{W}}^{cu}$ are simply connected, $\mathcal{W}^{cu}$ has no compact leaves.

Theorems & Definitions (51)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Definition 1.10
  • ...and 41 more