Propagation of regularity along unstable manifolds

TL;DR

This paper proves that regularity of certain dynamical states propagates along unstable manifolds, showing that smoothness on a piece implies smoothness everywhere, using advanced pseudodifferential calculus.

Contribution

It introduces a theorem of regularity propagation along unstable leaves for sections satisfying a transport equation, with applications to Pollicott-Ruelle resonant states and partially hyperbolic systems.

Findings

Regularity propagates along unstable manifolds for specific sections.

Smoothness on a subset implies global smoothness for Pollicott-Ruelle states.

Development of a leafwise semiclassical pseudodifferential calculus.

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.