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Propagation of regularity along unstable manifolds

Thibault Lefeuvre, Rafael Potrie

Abstract

Let $\varphi_t : M \to M$ be a flow on a smooth closed connected manifold $M$ that preserves and expands a foliation $F$. We establish a theorem of propagation of regularity along the leaves of $F$ for sections of vector bundles satisfying a transport equation involving the generator of a cocycle over $\varphi_t$. As a consequence, we prove a regularity result for Pollicott-Ruelle resonant states: if such state is smooth in restriction to a piece of an unstable leaf, then it is in fact smooth over the entire manifold. We also announce further applications related to joint integrability of extreme bundles of partially hyperbolic diffeomorphisms. The proofs rely on a leafwise semiclassical pseudodifferential calculus adapted to a foliated space, which may be of independent interest.

Propagation of regularity along unstable manifolds

Abstract

Let be a flow on a smooth closed connected manifold that preserves and expands a foliation . We establish a theorem of propagation of regularity along the leaves of for sections of vector bundles satisfying a transport equation involving the generator of a cocycle over . As a consequence, we prove a regularity result for Pollicott-Ruelle resonant states: if such state is smooth in restriction to a piece of an unstable leaf, then it is in fact smooth over the entire manifold. We also announce further applications related to joint integrability of extreme bundles of partially hyperbolic diffeomorphisms. The proofs rely on a leafwise semiclassical pseudodifferential calculus adapted to a foliated space, which may be of independent interest.
Paper Structure (50 sections, 26 theorems, 171 equations)

This paper contains 50 sections, 26 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Propagation of regularity
  3. Global smoothness
  4. Application to Pollicott-Ruelle resonant states
  5. Non joint integrability
  6. Leafwise semiclassical pseudodifferential calculus
  7. Comparison with other work
  8. Organization of the paper
  9. Conventions for distributions
  10. Leafwise pseudodifferential operators
  11. Foliations
  12. Definition
  13. Leafwise diffeomorphisms
  14. Tangent and cotangent bundles
  15. Leafwise smooth functions
  16. ...and 35 more sections

Key Result

Theorem 1.1

Assume that $E \to M$ is a leafwise smooth vector bundle. Let $\lambda \in \mathbb{C}$, $\gamma > 0$ and let $N \geq 0$ be an integer such that There exist constants $C := C(\mathbf{X}, N, \gamma) > 0$ and $\nu := \nu(\mathbf{X},N) > 0, \nu' := \nu'(\mathbf{X},N)>0$, such that the following holds. Let $u \in C^0(M,E)$ be a section such that $e^{t\mathbf{X}} u = e^{\lambda t} u$, and assume there

Theorems & Definitions (64)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Definition 2.1: Transversally continuous foliation with smooth leaves
  • Definition 2.2: Leafwise diffeomorphisms
  • Remark 2.3
  • Definition 2.4: Leafwise smooth functions
  • Definition 2.5: Leafwise smooth vector bundles
  • ...and 54 more