Radial source estimates in H\"older-Zygmund spaces for hyperbolic dynamics

TL;DR

This paper establishes a new radial source estimate in H"older-Zygmund spaces for hyperbolic dynamics, leading to improved stability results in the marked length spectrum rigidity conjecture for negatively curved metrics.

Contribution

It introduces a novel radial source estimate in H"older-Zygmund spaces for Anosov flows, enhancing stability results and regularity statements in hyperbolic dynamics.

Findings

Linear stability estimate for the marked length spectrum conjecture

Invariance of metrics with same spectrum in dimensions ≥ 2

Retrieval of regularity results in hyperbolic dynamics

Abstract

We prove a radial source estimate in H\"older-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 $\geq 2$, in the space of negatively-curved metrics, $C^{3+\varepsilon}$-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 Journ\'e's lemma: the smooth Liv\v{s}ic Theorem of de La Llave-Marco-Moriy\'on, the smooth Liv\v{s}ic cocycle theorem of Nitic\=a-T\"or\"ok for general (finite-dimensional) Lie groups, the rigidity of the regularity of the foliation obtained by Hasselblatt and others.