Abstract

We develop a paradifferential approach for studying nonsmooth hyperbolic dynamics on manifolds and related nonlinear PDEs from a microlocal point of view. As an application, we describe the microlocal regularity, i.e., the $H^s$ wavefront 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 the Hölder: 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). It is also shown in the Appendix that it can be applied to deal with nonsmooth flows and potentials. This work could serve as a toolbox for other applications.

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