Radial source estimates in Hölder-Zygmund spaces for hyperbolic dynamics

Guedes Bonthonneau, Yannick; Lefeuvre, Thibault

Annales Henri Lebesgue, Tome 6 (2023), pp. 643-686

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 [DZ16]. The main consequence is a new linear stability estimate for the marked length spectrum rigidity conjecture, also known as the Burns–Katok [BK85] 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 [GKL22, GL19]. 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 [LMM86], the smooth Livšic cocycle theorem of Niticā–Török [NT98] for general (finite-dimensional) Lie groups, the rigidity of the regularity of the foliation obtained by Hasselblatt [Has92] and others.

Nous établissons des estimées radiales de type source dans des espaces de régularité Hölder–Zygmund pour des dynamiques uniformément hyperboliques (flots Anosov), dans l’esprit des travaux de Dyatlov–Zworski [DZ16]. La principale conséquence est une nouvelle estimée de stabilité linéaire pour la conjecture de rigidité du spectre marqué des longueurs, aussi connue sous le nom de conjecture de Burns–Katok [BK85]. Nous montrons en particulier qu’en toute dimension $\ge 2$, dans l’espace des métriques à courbure négative, deux métriques $C^{3+\epsilon }$-proches avec même spectre marqué sont isométriques. Cela améliore des travaux récents de Guillarmou-Knieper et du second auteur [GKL22, GL19]. Cette approche permet aussi de retrouver divers résultats de régularité connus en dynamique hyperbolique et basés sur le lemme de Journé : le théorème de Livšic lisse de de La Llave–Marco–Moriyón [LMM86], la version cocycle du théorème de Livšic lisse de Niticā–Török [NT98] pour des groupes de Lie généraux (de dimension finie), la rigidité de la régularité du feuilletage obtenue par Hasselblatt [Has92] et d’autres.

Classification : 35P25, 35R30, 37D20, 37D40
Keywords: Radial source estimates, semiclassical analysis, hyperbolic dynamics, marked length spectrum rigidity

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