Vol. 14, No. 1, 2021

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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
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 Πm — 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