## The marked length spectrum of Anosov manifolds.(English)Zbl 07097501

Summary: In all dimensions, we prove that the marked length spectrum of a Riemannian manifold $$(M,g)$$ with Anosov geodesic flow and non-positive curvature locally determines the metric in the sense that two close enough metrics with the same marked length spectrum are isometric. In addition, we provide a new stability estimate quantifying how the marked length spectrum controls the distance between the isometry classes of metrics. In dimension $$2$$ we obtain similar results for general metrics with Anosov geodesic flows. We also solve locally a rigidity conjecture of Croke relating volume and marked length spectrum for the same category of metrics. Finally, by a compactness argument, we show that the set of negatively curved metrics (up to isometry) with the same marked length spectrum and with curvature in a bounded set of $$C^\infty$$ is finite.

### MSC:

 53C24 Rigidity results 53C22 Geodesics in global differential geometry 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 37C27 Periodic orbits of vector fields and flows
Full Text:

### References:

 [1] Anosov, D. V., Geodesic flows on closed {R}iemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov.. Akademiya Nauk SSSR. Trudy Matematicheskogo Instituta imeni V. A. Steklova, 90, 209 pp., (1967) [2] Besson, G.; Courtois, G.; Gallot, S., Entropies et rigidit\'{e}s des espaces localement sym\'{e}triques de courbure strictement n\'{e}gative, Geom. Funct. Anal.. Geometric and Functional Analysis, 5, 731-799, (1995) · Zbl 0851.53032 [3] Burns, K.; Katok, A., Manifolds with non-positive curvature, Ergodic Theory Dynam. Systems. Ergodic Theory and Dynamical Systems, 5, 307-317, (1985) · Zbl 0572.58019 [4] Butterley, Oliver; Liverani, Carlangelo, Smooth {A}nosov flows: correlation spectra and stability, Journal of Modern Dynamics, 1, 301-322, (2007) · Zbl 1144.37011 [5] Croke, Christopher B., Rigidity for surfaces of non-positive curvature, Comment. Math. Helv.. Commentarii Mathematici Helvetici, 65, 150-169, (1990) · Zbl 0704.53035 [6] Croke, Christopher B., Rigidity theorems in {R}iemannian geometry. Geometric Methods in Inverse Problems and {PDE} Control, IMA Vol. Math. Appl., 137, 47-72, (2004) · Zbl 1080.53033 [7] Croke, Christopher B.; Dairbekov, Nurlan S., Lengths and volumes in {R}iemannian manifolds, Duke Math. J.. Duke Mathematical Journal, 125, 1-14, (2004) · Zbl 1073.53053 [8] Croke, Christopher B.; Dairbekov, Nurlan S.; Sharafutdinov, Vladimir A., Local boundary rigidity of a compact {R}iemannian manifold with curvature bounded above, Trans. Amer. Math. Soc.. Transactions of the American Mathematical Society, 352, 3937-3956, (2000) · Zbl 0958.53027 [9] Croke, Christopher B.; Fathi, A.; Feldman, J., The marked length-spectrum of a surface of nonpositive curvature, Topology. Topology. An International Journal of Mathematics, 31, 847-855, (1992) · Zbl 0779.53025 [10] Croke, Christopher B.; Sharafutdinov, Vladimir A., Spectral rigidity of a compact negatively curved manifold, Topology. Topology. An International Journal of Mathematics, 37, 1265-1273, (1998) · Zbl 0936.58013 [11] Dairbekov, Nurlan S.; Sharafutdinov, Vladimir A., Some problems of integral geometry on {A}nosov manifolds, Ergodic Theory Dynam. Systems. Ergodic Theory and Dynamical Systems, 23, 59-74, (2003) · Zbl 1140.58302 [12] Dang, N. V.; Guillarmou, C.; Rivi\ere, G.; Shen, S., Fried conjecture in small dimensions, (2018) [13] de la Llave, R.; Marco, J. M.; Moriy\'{o}n, R., Canonical perturbation theory of {A}nosov systems and regularity results for the {L}iv\v{s}ic cohomology equation, Ann. of Math. (2). Annals of Mathematics. Second Series, 123, 537-611, (1986) · Zbl 0603.58016 [14] Appendix B. coauthored with H. Hezari, Dynamical spectral rigidity among {$$\Bbb Z_2$$}-symmetric strictly convex domains close to a circle, Ann. of Math. (2). Annals of Mathematics. Second Series, 186, 277-314, (2017) · Zbl 1377.37062 [15] Dyatlov, Semyon; Zworski, Maciej, Dynamical zeta functions for {A}nosov flows via microlocal analysis, Ann. Sci. \'{E}c. Norm. Sup\'{e}r. (4). Annales Scientifiques de l’\'{E}cole Normale Sup\'{e}rieure. Quatri\“‘eme S\'”’{e}rie, 49, 543-577, (2016) · Zbl 1369.37028 [16] Dyatlov, Semyon; Zworski, Maciej, Mathematical theory of scattering resonances, 200, (2019) · Zbl 1369.37037 [17] Ebin, David G., On the space of {R}iemannian metrics, Bull. Amer. Math. Soc.. Bulletin of the American Mathematical Society, 74, 1001-1003, (1968) · Zbl 0172.22905 [18] Farb, Benson; Margalit, Dan, A Primer on Mapping Class Groups, Princeton Math. Ser., 49, xiv+472 pp., (2011) · Zbl 1245.57002 [19] Faure, Fr\'{e}d\'{e}ric; Sj\"{o}strand, Johannes, Upper bound on the density of {R}uelle resonances for {A}nosov flows, Comm. Math. Phys.. Communications in Mathematical Physics, 308, 325-364, (2011) · Zbl 1260.37016 [20] Gromov, M., Hyperbolic groups. Essays in Group Theory, Math. Sci. Res. Inst. Publ., 8, 75-263, (1987) [21] Guillarmou, Colin, Invariant distributions and {X}-ray transform for {A}nosov flows, J. Differential Geom.. Journal of Differential Geometry, 105, 177-208, (2017) · Zbl 1372.37059 [22] Guillarmou, Colin, Lens rigidity for manifolds with hyperbolic trapped sets, J. Amer. Math. Soc.. Journal of the American Mathematical Society, 30, 561-599, (2017) · Zbl 1377.53098 [23] Guillemin, V.; Kazhdan, D., Some inverse spectral results for negatively curved {$$2$$}-manifolds, Topology. Topology. An International Journal of Mathematics, 19, 301-312, (1980) · Zbl 0465.58027 [24] Hamilton, Richard S., A compactness property for solutions of the {R}icci flow, Amer. J. Math.. American Journal of Mathematics, 117, 545-572, (1995) · Zbl 0840.53029 [25] Hamenst\"{a}dt, U., Cocycles, symplectic structures and intersection, Geom. Funct. Anal.. Geometric and Functional Analysis, 9, 90-140, (1999) · Zbl 0951.37007 [26] Katok, Anatole, Four applications of conformal equivalence to geometry and dynamics, Ergodic Theory Dynam. Systems. Ergodic Theory and Dynamical Systems, $$8^*$$, Charles Conley Memorial Issue, 139-152, (1988) · Zbl 0668.58042 [27] Klingenberg, Wilhelm, Riemannian manifolds with geodesic flow of {A}nosov type, Ann. of Math. (2). Annals of Mathematics. Second Series, 99, 1-13, (1974) · Zbl 0272.53025 [28] Klingenberg, Wilhelm, Riemannian Geometry, de Gruyter Stud. Math., 1, x+396 pp., (1982) · Zbl 0495.53036 [29] Knieper, Gerhard, New results on noncompact harmonic manifolds, Comment. Math. Helv.. Commentarii Mathematici Helvetici. A Journal of the Swiss Mathematical Society, 87, 669-703, (2012) · Zbl 1287.53056 [30] Lefeuvre, T., Local marked boundary rigidity under hyperbolic trapping assumptions, J. Geom Anal., (2019) [31] Liv\v{s}ic, A. N., Cohomology of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat.. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, 36, 1296-1320, (1972) · Zbl 0273.58013 [32] Lopes, A. O.; Thieullen, Ph., Sub-actions for {A}nosov flows, Ergodic Theory Dynam. Systems. Ergodic Theory and Dynamical Systems, 25, 605-628, (2005) · Zbl 1078.37021 [33] Otal, Jean-Pierre, Le spectre marqu\'{e} des longueurs des surfaces \`“‘a courbure n\'”’{e}gative, Ann. of Math. (2). Annals of Mathematics. Second Series, 131, 151-162, (1990) · Zbl 0699.58018 [34] Parry, William, Equilibrium states and weighted uniform distribution of closed orbits. Dynamical Systems, Lecture Notes in Math., 1342, 617-625, (1988) · Zbl 0667.58056 [35] Paternain, Gabriel P.; Salo, Mikko; Uhlmann, Gunther, Spectral rigidity and invariant distributions on {A}nosov surfaces, J. Differential Geom.. Journal of Differential Geometry, 98, 147-181, (2014) · Zbl 1304.37021 [36] Paternain, Gabriel P.; Salo, Mikko; Uhlmann, Gunther, Invariant distributions, {B}eurling transforms and tensor tomography in higher dimensions, Math. Ann.. Mathematische Annalen, 363, 305-362, (2015) · Zbl 1328.53099 [37] Petkov, Vesselin M.; Stoyanov, Luchezar N., Geometry of Reflecting Rays and Inverse Spectral Problems, Pure Appl. Math., vi+313 pp., (1992) · Zbl 0761.35077 [38] Petkov, Vesselin M.; Stoyanov, Luchezar N., Geometry of the Generalized Geodesic Flow and Inverse Spectral Problems, xv+410 pp., (2017) · Zbl 1354.35002 [39] Pestov, Leonid; Uhlmann, Gunther, Two dimensional compact simple {R}iemannian manifolds are boundary distance rigid, Ann. of Math. (2). Annals of Mathematics. Second Series, 161, 1093-1110, (2005) · Zbl 1076.53044 [40] Pollicott, Mark; Sharp, Richard, Livsic theorems, maximizing measures and the stable norm, Dyn. Syst.. Dynamical Systems. An International Journal, 19, 75-88, (2004) · Zbl 1057.37023 [41] Sarnak, Peter, Determinants of {L}aplacians; heights and finiteness. Analysis, et cetera, 601-622, (1990) [42] Sharafutdinov, V. A., Integral Geometry of Tensor Fields, Inverse and Ill-posed Problems Series, 271 pp., (1994) · Zbl 0883.53004 [43] Stefanov, P.; Uhlmann, G.; Vasy, A., Local and global boundary rigidity and the geodesic {X}-ray transform in the normal gauge, (2017) [44] Vign\'{e}ras, Marie-France, Vari\'{e}t\'{e}s riemanniennes isospectrales et non isom\'{e}triques, Ann. of Math. (2). Annals of Mathematics. Second Series, 112, 21-32, (1980) · Zbl 0445.53026 [45] Wilkinson, Amie, Lectures on marked length spectrum rigidity. Geometric {G}roup {T}heory, IAS/Park City Math. Ser., 21, 283-324, (2014)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.