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Geodesic stretch, pressure metric and marked length spectrum rigidity

Part of: Smooth dynamical systems: general theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  09 August 2021

COLIN GUILLARMOU Open the ORCID record for COLIN GUILLARMOU [Opens in a new window] ,
GERHARD KNIEPER  and
THIBAULT LEFEUVRE Open the ORCID record for THIBAULT LEFEUVRE [Opens in a new window]
COLIN GUILLARMOU*
Affiliation:
Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405Orsay, France
GERHARD KNIEPER
Affiliation:
Ruhr-Universität Bochum, Fakultät für Mathematik, D-44780Bochum, Deutschland (e-mail: gerhard.knieper@rub.de)
THIBAULT LEFEUVRE
Affiliation:
Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405Orsay, France (e-mail: tlefeuvre@imj-prg.fr)
*
e-mail: colin.guillarmou@math.u-psud.fr
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Abstract

We refine the recent local rigidity result for the marked length spectrum obtained by the first and third author in [GL19] and give an alternative proof using the geodesic stretch between two Anosov flows and some uniform estimate on the variance appearing in the central limit theorem for Anosov geodesic flows. In turn, we also introduce a new pressure metric on the space of isometry classes, which reduces to the Weil–Petersson metric in the case of Teichmüller space and is related to the works [BCLS15, MM08].

Keywords

Anosov geodesic flowsrigiditymarked length spectrum

MSC classification

Primary: 37C27: Periodic orbits of vector fields and flows 37D35: Thermodynamic formalism, variational principles, equilibrium states 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 3: Anatole Katok Memorial Issue Part 2: Special Issue of Ergodic Theory and Dynamical Systems , March 2022 , pp. 974 - 1022
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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