Classical and microlocal analysis of the x-ray transform on Anosov manifolds

Gouëzel, Sébastien; Lefeuvre, Thibault

doi:10.2140/apde.2021.14.301

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Classical and microlocal analysis of the x-ray transform on Anosov manifolds

Sébastien Gouëzel and Thibault Lefeuvre

Vol. 14 (2021), No. 1, 301–322

DOI: 10.2140/apde.2021.14.301

Abstract

We complete the microlocal study of the geodesic x-ray transform on Riemannian manifolds with Anosov geodesic flow initiated by Guillarmou (J. Differential Geom. 105:2 (2017), 177–208) and pursued by Guillarmou and Lefeuvre in (Ann. of Math. $(2)$ 190:1 (2019), 321–344). We prove new stability estimates and clarify some properties of the operator $Π_{m}$ — the generalized x-ray transform. These estimates rely on a refined version of the Livšic theorem for Anosov flows, especially on a new quantitative finite-time Livšic theorem.

Keywords

Anosov flow, hyperbolic dynamical systems, x-ray transform, microlocal analysis

Mathematical Subject Classification 2010

Primary: 37C27, 37D40, 53C21, 53C22, 53C24

Milestones

Received: 17 May 2019

Revised: 9 August 2019

Accepted: 7 October 2019

Published: 19 February 2021

Authors

Sébastien Gouëzel
Laboratoire Jean Leray, CNRS UMR 6629 Université de Nantes Nantes France

Thibault Lefeuvre
Laboratoire de Mathématiques d’Orsay Université Paris-Sud, CNRS Université Paris-Saclay Orsay France

Vol. 14, No. 1, 2021