Radial source estimates in Hölder-Zygmund spaces for hyperbolic dynamics

Abstract

We prove a radial source estimate in Hölder-Zygmund spaces for uniformly hyperbolic dynamics (also known as Anosov flows), in the spirit of Dyatlov-Zworski. The main consequence is a new linear stability estimate for the marked length spectrum rigidity conjecture, also known as the Burns-Katok conjecture. We show in particular that in any dimension $\ge 2$, in the space of negatively-curved metrics, $C^{3+\epsilon }$-close metrics with same marked length spectrum are isometric. This improves recent works of Guillarmou-Knieper and the second author. As a byproduct, this approach also allows to retrieve various regularity statements known in hyperbolic dynamics and usually based on Journé's lemma: the smooth Livšic Theorem of de La Llave-Marco-Moriyón, the smooth Livšic cocycle theorem of Niticā-Török for general (finite-dimensional) Lie groups, the rigidity of the regularity of the foliation obtained by Hasselblatt and others.

46 pages, 1 figure