Volume 6 (2023)
Guedes Bonthonneau, Yannick; Lefeuvre, Thibault
Radial source estimates in Hölder-Zygmund spaces for hyperbolic dynamics
Annales Henri Lebesgue, Volume 6 (2023), pp. 643-686.

Metadata

DOI10.5802/ahl.175
Keywords Radial source estimates, semiclassical analysis, hyperbolic dynamics, marked length spectrum rigidity

Abstract

We prove a radial source estimate in Hölder–Zygmund spaces for uniformly hyperbolic dynamics (also known as Anosov flows), in the spirit of Dyatlov–Zworski [DZ16]. The main consequence is a new linear stability estimate for the marked length spectrum rigidity conjecture, also known as the Burns–Katok [BK85] conjecture. We show in particular that in any dimension ≥2, in the space of negatively-curved metrics, C 3+ε -close metrics with same marked length spectrum are isometric. This improves recent works of Guillarmou–Knieper and the second author [GKL22, GL19]. As a byproduct, this approach also allows to retrieve various regularity statements known in hyperbolic dynamics and usually based on Journé’s lemma: the smooth Livšic Theorem of de La Llave–Marco–Moriyón [LMM86], the smooth Livšic cocycle theorem of Niticā–Török [NT98] for general (finite-dimensional) Lie groups, the rigidity of the regularity of the foliation obtained by Hasselblatt [Has92] and others.


References

[AB22] Adam, Alexander; Baladi, Viviane Horocycle averages on closed manifolds and transfer operators, Tunis. J. Math., Volume 4 (2022) no. 3, pp. 387-441 | DOI | MR | Zbl

[Ada19] Adam, Alexander Horocycle averages on closed manifolds and transfer operators (2019) (https://arxiv.org/abs/1809.04062v2)

[BCG95] Besson, Gérard; Courtois, Gilles; Gallot, Sylvestre Entropies et rigidités des espaces localement symétriques de courbure strictement négative, Geom. Funct. Anal., Volume 5 (1995) no. 5, pp. 731-799 | DOI | MR | Zbl

[BFL90] Benoist, Yves; Foulon, Patrick; Labourie, François Flots d’Anosov à distributions de Liapounov différentiables. I, Ann. Inst. Henri Poincaré, Phys. Théor., Volume 53 (1990) no. 4, pp. 395-412 | Numdam | MR | Zbl

[BFL92] Benoist, Yves; Foulon, Patrick; Labourie, François Flots d’Anosov à distributions stable et instable différentiables, J. Am. Math. Soc., Volume 5 (1992) no. 1, pp. 33-74 | DOI | MR | Zbl

[BK85] Burns, Keith; Katok, Anatole Manifolds with nonpositive curvature, Ergodic Theory Dyn. Syst., Volume 5 (1985) no. 2, pp. 307-317 | DOI | MR | Zbl

[BT07] Baladi, Viviane; Tsujii, Masato Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms, Ann. Inst. Fourier, Volume 57 (2007) no. 1, pp. 127-154 | DOI | Numdam | MR | Zbl

[BT08] Baladi, Viviane; Tsujii, Masato Dynamical Determinants and Spectrum for Hyperbolic Diffemorphisms, Geometric and probabilistic structures in dynamics (Contemporary Mathematics), Volume 469, American Mathematical Society, 2008, pp. 29-68 | DOI | Zbl

[CdV20] Colin de Verdière, Yves Spectral theory of pseudodifferential operators of degree 0 and an application to forced linear waves, Anal. PDE, Volume 13 (2020) no. 5, pp. 1521-1537 | DOI | Zbl

[CL21] Cekić, Mihajlo; Lefeuvre, Thibault Generic injectivity of the X-ray transform (2021) (in preparation)

[CLa] Cekić, Mihajlo; Lefeuvre, Thibault The holonomy inverse problem on Anosov manifolds (in preparation)

[CLb] Cekić, Mihajlo; Lefeuvre, Thibault On transparent manifolds (in preparation) | DOI | MR | Zbl

[Cro90] Croke, Christopher B. Rigidity for surfaces of nonpositive curvature, Comment. Math. Helv., Volume 65 (1990) no. 1, pp. 150-169 | DOI | MR | Zbl

[CS98] Croke, Christopher B.; Sharafutdinov, Vladimir A. Spectral rigidity of a compact negatively curved manifold, Topology, Volume 37 (1998) no. 6, pp. 1265-1273 | DOI | MR | Zbl

[DD13] Datchev, Kiril; Dyatlov, Semyon Fractal Weyl laws for asymptotically hyperbolic manifolds, Geom. Funct. Anal., Volume 23 (2013) no. 4, pp. 1145-1206 | DOI | MR | Zbl

[deLl99] de la Llave, Rafael; Obaya, Rafael Regularity Of The Composition Operator In Spaces Of Hölder Functions, Discrete Contin. Dyn. Syst., Volume 5 (1999), pp. 157-184 | DOI | MR | Zbl

[deLl01] de la Llave, Rafael Remarks on Sobolev regularity in Anosov systems, Ergodic Theory Dyn. Syst., Volume 21 (2001) no. 4, pp. 1139-1180 | DOI | MR | Zbl

[DH72] Duistermaat, Johannes J.; Hörmander, Lars Fourier integral operators. II, Acta Math., Volume 128 (1972) no. 3-4, pp. 183-269 | DOI | MR | Zbl

[DS99] Dimassi, Mouez; Sjöstrand, Johannes Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, 268, Cambridge University Press, 1999 | DOI | MR | Zbl

[Dya12] Dyatlov, Semyon Asymptotic distribution of quasi-normal modes for Kerr–de Sitter black holes, Ann. Henri Poincaré, Volume 13 (2012) no. 5, pp. 1101-1166 | DOI | MR | Zbl

[DZ16] Dyatlov, Semyon; Zworski, Maciej Dynamical zeta functions for Anosov flows via microlocal analysis, Ann. Sci. Éc. Norm. Supér., Volume 49 (2016) no. 3, pp. 543-577 | DOI | MR | Zbl

[DZ19a] Dyatlov, Semyon; Zworski, Maciej Mathematical theory of scattering resonances, Graduate Studies in Mathematics, 200, American Mathematical Society, 2019 | DOI | MR | Zbl

[DZ19b] Dyatlov, Semyon; Zworski, Maciej Microlocal analysis of forced waves, Pure Appl. Anal., Volume 1 (2019) no. 3, pp. 359-384 | DOI | MR | Zbl

[Ebi68] Ebin, David G. On the space of Riemannian metrics, Bull. Am. Math. Soc., Volume 74 (1968), pp. 1001-1003 | DOI | MR | Zbl

[FS11] Faure, Frédéric; Sjöstrand, Johannes Upper bound on the density of Ruelle resonances for Anosov flows, Commun. Math. Phys., Volume 308 (2011) no. 2, pp. 325-364 | DOI | MR | Zbl

[FT13] Faure, Frédéric; Tsujii, Masato Band structure of the Ruelle spectrum of contact Anosov flows, C. R. Math. Acad. Sci. Paris, Volume 351 (2013) no. 9-10, pp. 385-391 | DOI | Numdam | MR | Zbl

[GDP] Guillarmou, Colin; De Poyferré, Thibault A paradifferential approach for hyperbolic dynamical systems and applications (In preparation)

[GKL22] Guillarmou, Colin; Knieper, Gerhard; Lefeuvre, Thibault Geodesic stretch, pressure metric and marked length spectrum rigidity, Ergodic Theory Dyn. Syst., Volume 42 (2022) no. 3, pp. 974-1022 | DOI | MR | Zbl

[GL19] Guillarmou, Colin; Lefeuvre, Thibault The marked length spectrum of Anosov manifolds, Ann. Math., Volume 190 (2019) no. 1, pp. 321-344 | DOI | MR | Zbl

[GL21] Gouëzel, Sébastien; Lefeuvre, Thibault Classical and microlocal analysis of the x-ray transform on Anosov manifolds, Anal. PDE, Volume 14 (2021) no. 1, pp. 301-322 | DOI | MR | Zbl

[GLP13] Giulietti, Paolo; Liverani, Carlangelo; Pollicott, Mark Anosov flows and dynamical zeta functions, Ann. Math., Volume 178 (2013) no. 2, pp. 687-773 | DOI | MR | Zbl

[Gui17] Guillarmou, Colin Invariant distributions and X-ray transform for Anosov flows, J. Differ. Geom., Volume 105 (2017) no. 2, pp. 177-208 | MR | Zbl

[Hal15] Hall, Brian Lie groups, Lie algebras, and representations, Graduate Texts in Mathematics, 222, Springer, 2015 (An elementary introduction) | DOI | MR | Zbl

[Ham99] Hamenstädt, Ursula Cocycles, symplectic structures and intersection, Geom. Funct. Anal., Volume 9 (1999) no. 1, pp. 90-140 | DOI | MR | Zbl

[Has92] Hasselblatt, Boris Bootstrapping regularity of the Anosov splitting, Proc. Am. Math. Soc., Volume 115 (1992) no. 3, pp. 817-819 | DOI | MR | Zbl

[HK90] Hurder, Steven; Katok, Anatole Differentiability, rigidity and Godbillon–Vey classes for Anosov flows, Publ. Math., Inst. Hautes Étud. Sci. (1990) no. 72, pp. 5-61 | DOI | Numdam | MR | Zbl

[HMV04] Hassell, Andrew; Melrose, Richard B.; Vasy, András Spectral and scattering theory for symbolic potentials of order zero, Adv. Math., Volume 181 (2004) no. 1, pp. 1-87 | DOI | MR | Zbl

[HV18] Hintz, Peter; Vasy, András The global non-linear stability of the Kerr–de Sitter family of black holes, Acta Math., Volume 220 (2018) no. 1, pp. 1-206 | DOI | MR | Zbl

[Jou86] Journé, Jean-Lin On a regularity problem occurring in connection with Anosov diffeomorphisms, Commun. Math. Phys., Volume 106 (1986) no. 2, pp. 345-351 | DOI | MR | Zbl

[Kat88] Katok, Anatole Four applications of conformal equivalence to geometry and dynamics, Ergodic Theory Dyn. Syst., Volume 8 * (1988) no. Charles Conley Memorial Issue, pp. 139-152 | DOI | MR | Zbl

[KH95] Katok, Anatole; Hasselblatt, Boris Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and Its Applications, 54, Cambridge University Press, 1995 (With a supplementary chapter by Katok and Leonardo Mendoza) | DOI | MR | Zbl

[KKPW89] Katok, Anatole; Knieper, Gerhard; Pollicott, Mark; Weiss, Howard Differentiability and analyticity of topological entropy for Anosov and geodesic flows, Invent. Math., Volume 98 (1989), pp. 581-597 | DOI | MR | Zbl

[Liv72] Livšic, A. N. Cohomology of dynamical systems, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 36 (1972), pp. 1296-1320 | MR

[LMM86] de la Llave, Rafael; Marco, José M.; Moriyón, Roberto Canonical perturbation theory of Anosov systems and regularity results for the Livšic cohomology equation, Ann. Math., Volume 123 (1986) no. 3, pp. 537-611 | DOI | MR | Zbl

[LT05] Lopes, Artur O.; Thieullen, Philippe Sub-actions for Anosov flows, Ergodic Theory Dyn. Syst., Volume 25 (2005) no. 2, pp. 605-628 | DOI | MR | Zbl

[Mel94] Melrose, Richard B. Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces, Spectral and scattering theory (Sanda, 1992) (Lecture Notes in Pure and Applied Mathematics), Volume 161, Marcel Dekker, 1994, pp. 85-130 | MR | Zbl

[NT98] Niţică, Viorel; Török, Andrei Regularity of the transfer map for cohomologous cocycles, Ergodic Theory Dyn. Syst., Volume 18 (1998) no. 5, pp. 1187-1209 | DOI | MR | Zbl

[Ota90] Otal, Jean-Pierre Le spectre marqué des longueurs des surfaces à courbure négative, Ann. Math., Volume 131 (1990) no. 1, pp. 151-162 | DOI | MR | Zbl

[Pat99] Paternain, Gabriel P. Geodesic flows, Progress in Mathematics, 180, Birkhäuser, 1999 | DOI | MR | Zbl

[PSU14] Paternain, Gabriel P.; Salo, Mikko; Uhlmann, Gunther Tensor tomography: progress and challenges, Chin. Ann. Math., Ser. B, Volume 35 (2014) no. 3, pp. 399-428 | DOI | MR | Zbl

[RS96] Runst, Thomas; Sickel, Winfried Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, de Gruyter Series in Nonlinear Analysis and Applications, 3, Walter de Gruyter, 1996 | DOI | MR | Zbl

[Vas13] Vasy, András Microlocal analysis of asymptotically hyperbolic and Kerr–de Sitter spaces (with an appendix by Semyon Dyatlov), Invent. Math., Volume 194 (2013) no. 2, pp. 381-513 | DOI | MR | Zbl

[Wan20] Wang, Jian Sharp radial estimates in Besov spaces (2020) (https://arxiv.org/abs/2003.11218)

[Zwo12] Zworski, Maciej Semiclassical analysis, Graduate Studies in Mathematics, 138, American Mathematical Society, 2012 | DOI | MR | Zbl