A paradifferential approach for hyperbolic dynamical systems and applications

TL;DR

This paper introduces a paradifferential microlocal approach to analyze non-smooth hyperbolic dynamical systems and PDEs, providing new insights into the regularity of unstable bundles and rigidity phenomena.

Contribution

It develops a novel paradifferential method for studying non-smooth hyperbolic dynamics and applies it to establish Sobolev regularity results and rigidity theorems for Anosov flows.

Findings

Describes the $H^s$ wave-front set of the unstable bundle for all $s$.

Shows that $H^s$ regularity implies smoothness for $s > s_0$, with $s_0=2$ in certain cases.

Applicable to non-smooth flows and potentials, broadening the scope of analysis.

Abstract

We develop a paradifferential approach for studying non-smooth hyperbolic dynamics and related non-linear PDE from a microlocal point of view. As an application, we describe the microlocal regularity, i.e the $H^s$ wave-front set for all $s$, of the unstable bundle $E_u$ for an Anosov flow. We also recover rigidity results of Hurder-Katok and Hasselblatt in the Sobolev class rather than H\"older: there is $s_0>0$ such that if $E_u$ has $H^s$ regularity for $s>s_0$ then it is smooth (with $s_0=2$ for volume preserving $3$-dimensional Anosov flows). In the appendix by Guedes Bonthonneau, it is also shown that it can be applied to deal with non-smooth flows and potentials. This work could serve as a toolbox for other applications.