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A paradifferential approach for hyperbolic dynamical systems and applications

Colin Guillarmou and Thibault de Poyferré

Appendix: Yannick Guedes Bonthonneau

Vol. 4 (2022), No. 4, 673–718

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 Hs wavefront set for all s, of the unstable bundle Eu for an Anosov flow. We also recover rigidity results of Hurder–Katok and Hasselblatt in the Sobolev class rather than the Hölder: there is s0 > 0 such that if Eu has Hs regularity for s > s0, then it is smooth (with s0 = 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.

hyperbolic dynamics, paradifferential calculus, regularity of foliations, Ruelle resonances
Mathematical Subject Classification
Primary: 35B65, 58J47, 58Jxx
Secondary: 37D20
Received: 1 November 2021
Revised: 15 June 2022
Accepted: 29 June 2022
Published: 15 January 2023
Colin Guillarmou
Laboratoire de Mathématiques d’Orsay
Université Paris-Saclay
Thibault de Poyferré
MSRI Berkeley
Berkeley, CA
United States
Yannick Guedes Bonthonneau
Institut Galilée
Université Paris 13