Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-25T12:08:53.483Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Avila, A., De Simoi, J. and Kaloshin, V.. An integrable deformation of an ellipse of small eccentricity is an ellipse. Ann. of Math. 184 (2016), 527–558.10.4007/annals.2016.184.2.5CrossRefGoogle Scholar
Besson, G., Courtois, G. and Gallot, S.. Entropies et rigidités des espaces localement symétriques de courbure strictement négative. Geom. Funct. Anal. 5(5) (1995), 731–799.10.1007/BF01897050CrossRefGoogle Scholar
Bridgeman, M., Canary, R., Labourie, F. and Sambarino, A.. The pressure metric for Anosov representations. Geom. Funct. Anal. 25(4) (2015), 1089–1179.10.1007/s00039-015-0333-8CrossRefGoogle Scholar
Bridgeman, M., Canary, R. and Sambarino, A.. An introduction to pressure metrics for higher Teichmüller spaces. Ergod. Th. & Dynam. Sys. 38(6) (2018), 2001–2035.10.1017/etds.2016.111CrossRefGoogle Scholar
Bridson, M. R. and Haefliger, A.. Metric Spaces of Non-Positive Curvature (Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319). Springer, Berlin, 1999.Google Scholar
Burns, K. and Katok, A.. Manifolds with nonpositive curvature. Ergod. Th. & Dynam. Sys. 5(2) (1985), 307–317.10.1017/S0143385700002935CrossRefGoogle Scholar
Bonthonneau, Y. G.. Flow-independent Anisotropic space, and perturbation of resonances. Rev. Un. Mat. Argentina 61(1) (2020), 63–72.10.33044/revuma.v61n1a03CrossRefGoogle Scholar
Bowen, R. and Ruelle, D.. The ergodic theory of Axiom A flows. Invent. Math. 29(3) (1975), 181–202.10.1007/BF01389848CrossRefGoogle Scholar
Croke, C. B. and Fathi, A.. An inequality between energy and intersection. Bull. Lond. Math. Soc. 22 (1990), 489–494.CrossRefGoogle Scholar
Contreras, G.. Regularity of topological and metric entropy of hyperbolic flows. Math. Z. 210(1) (1992), 97–111.10.1007/BF02571785CrossRefGoogle Scholar
Croke, C. B.. Rigidity for surfaces of nonpositive curvature. Comment. Math. Helv. 65(1) (1990), 150–169.10.1007/BF02566599CrossRefGoogle Scholar
Croke, C. B. and Sharafutdinov, V. A.. Spectral rigidity of a compact negatively curved manifold. Topology 37(6) (1998), 1265–1273.10.1016/S0040-9383(97)00086-4CrossRefGoogle Scholar
Dang, N. V., Guillarmou, C., Rivière, G. and Shen, S.. Fried conjecture in small dimensions. Invent. Math. 220 (2020), 525–579.10.1007/s00222-019-00935-9CrossRefGoogle Scholar
de la Llave, R., Marco, J. M. and Moriyón, R.. Canonical perturbation theory of Anosov systems and regularity results for the Livšic cohomology equation. Ann. of Math. (2) 123(3) (1986), 537–611.10.2307/1971334CrossRefGoogle Scholar
Dairbekov, N. S. and Sharafutdinov, V. A.. Some problems of integral geometry on Anosov manifolds. Ergod. Th. & Dynam. Sys. 23 (2003), 59–74.10.1017/S0143385702000822CrossRefGoogle Scholar
Dairbekov, N. S. and Sharafutdinov, V. A.. Conformal Killing symmetric tensor fields on Riemannian manifolds. Mat. Tr. 13(1) (2010), 85–145.Google Scholar
De Simoi, J., Kaloshin, V. and Wei, Q.. Dynamical spectral rigidity among $2$ -symmetric strictly convex domains close to a circle. Ann. of Math. (2) 186 (2017), 277–314.10.4007/annals.2017.186.1.7CrossRefGoogle Scholar
Dyatlov, S. and Zworski, M.. Dynamical zeta functions for Anosov flows via microlocal analysis. Ann. Sci. Éc. Norm. Supér. (4) 49(3) (2016), 543–577.10.24033/asens.2290CrossRefGoogle Scholar
Dyatlov, S. and Zworski, M.. Mathematical Theory of Scattering Resonances (Graduate Studies in Mathematics, 200). American Mathematical Society, Providence, RI, 2019.10.1090/gsm/200CrossRefGoogle Scholar
Ebin, D. G.. On the space of Riemannian metrics. Bull. Amer. Math. Soc. 74 (1968), 1001–1003.10.1090/S0002-9904-1968-12115-9CrossRefGoogle Scholar
Ebin, D. G.. The manifold of Riemannian metrics. Global Analysis (Proceedings of Symposia in Pure Mathematics, 15). Eds. Chern, S.-S. and Smale, S.. American Mathematical Society, Providence, RI, 1970, pp. 11–40.Google Scholar
Fathi, A. and Flaminio, L.. Infinitesimal conjugacies and Weil–Petersson metric. Ann. Inst. Fourier 43(1) (1993), 279–299.10.5802/aif.1331CrossRefGoogle Scholar
Flaminio, L.. Local entropy rigidity for hyperbolic manifolds. Comm. Anal. Geom. 3(4) (1995), 555–596.CrossRefGoogle Scholar
Frankel, T.. On theorems of Hurwitz and Bochner. J. Math. Mech. 15 (1966), 373–377.Google Scholar
Faure, F., Roy, N. and Sjöstrand, J.. Semi-classical approach for Anosov diffeomorphisms and Ruelle resonances. Open J. Math. 1 (2008), 35–81.CrossRefGoogle Scholar
Faure, F. and Sjöstrand, J.. Upper bound on the density of Ruelle resonances for Anosov flows. Comm. Math. Phys. 308(2) (2011), 325–364.CrossRefGoogle Scholar
Giulietti, P., Kloeckner, B., Lopes, A. O. and Marcon, D.. The calculus of thermodynamical formalism. J. Eur. Math. Soc. (JEMS) 20(10) (2018), 2357–2412.CrossRefGoogle Scholar
Goüezel, S. and Lefeuvre, T.. Classical and microlocal analysis of the X-ray transform on Anosov manifolds. Anal. PDE 14(1) (2021), 301–322.CrossRefGoogle Scholar
Guillarmou, C. and Lefeuvre, T.. The marked length spectrum of Anosov manifolds. Ann. of Math. (2) 190(1) (2019), 321–344.CrossRefGoogle Scholar
Gromov, M.. Three remarks on geodesic dynamics and fundamental group. Enseign. Math. (2) 46(3–4) (2000), 391–402.Google Scholar
Grigis, A. and Sjöstrand, J.. Microlocal Analysis for Differential Operators: An Introduction. Cambridge University Press, Cambridge, 1994.10.1017/CBO9780511721441CrossRefGoogle Scholar
Guillarmou, C.. Invariant distributions and X-ray transform for Anosov flows. J. Differential Geom. 105(2) (2017), 177–208.CrossRefGoogle Scholar
Hamilton, R. S.. The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. 7(1) (1982), 65–222.CrossRefGoogle Scholar
Hamenstädt, U.. Cocycles, symplectic structures and intersection. Geom. Funct. Anal. 9(1) (1999), 90–140.CrossRefGoogle Scholar
Hasselblatt, B. and Fisher, T.. Hyperbolic Flows (Zurich Lectures in Advanced Mathematics). European Mathematical Society, Zurich, 2019.Google Scholar
Hurder, S. and Katok, A.. Differentiability, rigidity and Godbillon–Vey classes for Anosov flows. Publ. Math. Inst. Hautes Études Sci. 72 (1990), 5–61 (1991).CrossRefGoogle Scholar
Hörmander, L.. The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis, reprint of the 2nd (1990) edition. Springer, Berlin, 2003.Google Scholar
Hirsch, M. W. and Pugh, C. C.. Stable manifolds and hyperbolic sets. Global Analysis (Proceedings of Symposia in Pure Mathematics, XIV, Berkeley, CA, 1968). Ed. Chern, S. and Smale, S.. American Mathematical Society, Providence, RI, pp. 133–163.Google Scholar
Katok, A.. Four applications of conformal equivalence to geometry and dynamics. Ergod. Th. & Dynam. Sys. 8(8: Charles Conley Memorial Issue) (1988), 139–152.Google Scholar
Katok, A., Knieper, G., Pollicott, M. and Weiss, H.. Differentiability and analyticity of topological entropy for Anosov and geodesic flows. Invent. Math. 98 (1989), 581–597.CrossRefGoogle Scholar
Katok, A., Knieper, G., Pollicott, M. and Weiss, H.. Differentiability of entropy for Anosov and geodesic flows. Bull. Amer. Math. Soc. (N.S.) 22(2) (1990), 285–293.10.1090/S0273-0979-1990-15889-6CrossRefGoogle Scholar
Katok, A., Knieper, G. and Weiss, H.. Formulas for the derivative and critical points of topological entropy for Anosov and geodesic flows. Comm. Math. Phys. 138(1) (1991), 19–31.CrossRefGoogle Scholar
Klingenberg, W.. Riemannian manifolds with geodesic flow of Anosov type. Ann. of Math. (2) 99 (1974), 1–13.CrossRefGoogle Scholar
Knieper, G.. Hyperbolic dynamical systems. Handbook of Dynamical Systems. Vol. 1A. Eds. Hasselblat, B. and Katok, A.. Elsevier, Amsterdam, 2002, pp. 239–319.Google Scholar
Knieper, G.. New results on noncompact harmonic manifolds. Comment. Math. Helv. 87(3) (2012), 669–703.10.4171/CMH/265CrossRefGoogle Scholar
Knieper, G.. Volume growth, entropy and the geodesic stretch. Math. Res. Lett. 2(1) (1995), 39–58.10.4310/MRL.1995.v2.n1.a5CrossRefGoogle Scholar
Katsuda, A. and Sunada, T.. Closed orbits in homology classes. Publ. Math. Inst. Hautes Études Sci. 71 (1990), 5–32.10.1007/BF02699875CrossRefGoogle Scholar
Liverani, C.. On contact Anosov flows. Ann. of Math. (2) 159(3) (2004), 1275–1312.CrossRefGoogle Scholar
Lopes, A.O. and Ruggiero, R. O.. The sectional curvature of the infinite dimensional manifold of Hölder equilibirum probabilities. Preprint, 2018, arXiv:1811.07748.Google Scholar
Melrose, R. B.. Differential analysis on manifolds with corners, book in preparation, available at http://www-math.mit.edu/~rbm/book.html.Google Scholar
McMullen, C. T.. Thermodynamics, dimension and the Weil–Petersson metric. Invent. Math. 173(2) (2008), 365–425.10.1007/s00222-008-0121-2CrossRefGoogle Scholar
Otal, J.-P.. Le spectre marqué des longueurs des surfaces à courbure négative. Ann. of Math. (2) 131(1) (1990), 151–162.CrossRefGoogle Scholar
Parry, W.. Equilibrium states and weighted uniform distribution of closed orbits. Dynamical Systems (Lecture Notes in Mathematics, 1342). Ed. Alexander, J. C.. Spinger, Berlin, 1988, pp. 617–625.Google Scholar
Paternain, G. P.. Geodesic Flows (Progress in Mathematics, 180). Birkhäuser, Boston, 1999.10.1007/978-1-4612-1600-1CrossRefGoogle Scholar
Pollicott, M.. Derivatives of topological entropy for Anosov and geodesic flows. J. Differential Geom. 39 (1994), 457–489.10.4310/jdg/1214455077CrossRefGoogle Scholar
Parry, W. and Pollicott, M.. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187–188 (1990), 268pp.Google Scholar
Paulin, F., Pollicott, M. and Schapira, B.. Equilibrium states in negative curvature. Astérisque 373 (2015), viii+281pp.Google Scholar
Paternain, G. P., Salo, M. and Uhlmann, G.. Spectral rigidity and invariant distributions on Anosov surfaces. J. Differential Geom. 98(1) (2014), 147–181.CrossRefGoogle Scholar
Ruelle, D.. Thermodynamic formalism. The Mathematical Structures of Equilibrium Statistical Mechanics, 2nd edn. Cambridge University Press, Cambridge, 2004.CrossRefGoogle Scholar
Sambarino, A.. Quantitative properties of convex representations. Comment. Math. Helv. 89(2) (2014), 443–488.CrossRefGoogle Scholar
Sigmund, K.. On the space of invariant measures for hyperbolic flows. Amer. J. Math. 94(1) (1972), 31–37.CrossRefGoogle Scholar
Sharafutdinov, V., Skokan, S. and Uhlmann, G.. Regularity of ghosts in tensor tomography. J. Geom. Anal. 15(3) (2005), 499–542.CrossRefGoogle Scholar
Schapira, B. and Tapie, S.. Regularity of entropy, geodesic currents and entropy at infinity. Ann. Sci. Éc. Norm. Supér. (4), to appear.Google Scholar
Stefanov, P. and Uhlmann, G.. Stability estimates for the X-ray transform of tensor fields and boundary rigidity. Duke Math. J. 123(3) (2004), 445–467.CrossRefGoogle Scholar
Taylor, M. E.. Pseudodifferential Operators and Nonlinear PDE (Progress in Mathematics, 100). Birkhäuser, Boston, 1991.CrossRefGoogle Scholar
Taylor, M. E.. Partial Differential Equations: Basic Theory (Applied Mathematical Sciences, 115). Springer, New York, 1996.CrossRefGoogle Scholar
Thurston, W. P.. Minimal stretch maps between hyperbolic surfaces. Preprint, 1998, arXiv:math/9801039.Google Scholar
Tromba, A. J.. Teichmüller Theory in Riemannian Geometry (Lectures in Mathematics ETH Zürich). Birkhäuser, Basel, 1992. Lecture notes prepared by Jochen Denzler.CrossRefGoogle Scholar
Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York and Berlin, 1982.CrossRefGoogle Scholar
Wolpert, S.. Thurston’s Riemannian metric for Teichmüller space. J. Differential Geom. 23 (1986), 143–174.CrossRefGoogle Scholar
Zeidler, E.. Nonlinear Functional Analysis and Its Applications. IV: Applications to Mathematical Physics. Springer, New York, 1988. Translated from the German and with a preface by Juergen Quandt.CrossRefGoogle Scholar