Fractal Weyl law for the Ruelle spectrum of Anosov flows
Annales Henri Lebesgue, Volume 6 (2023), pp. 331-426.


Keywords Transfer operator, Ruelle resonances, decay of correlations, Semi-classical analysis


On a closed manifold M, we consider a smooth vector field X that generates an Anosov flow. Let VC (M;) be a smooth function called potential. It is known that for any C>0, there exists some anisotropic Sobolev space C such that the operator A=-X+V has intrinsic discrete spectrum on Re(z)>-C called Ruelle resonances. In this paper, we show a “Fractal Weyl law”: the density of resonances is bounded by O(ω n 1+β 0 ) where ω=Im(z), n=dimM-1 and 0<β 0 1 is the Hölder exponent of the distribution E u E s (strong stable and unstable). We also obtain some more precise results concerning the wave front set of the resonances and the invertibility of the transfer operator. Since the dynamical distributions E u ,E s are non smooth, we use some semi-classical analysis based on wave packet transform associated to an adapted metric g on T * M and construct some specific anisotropic Sobolev spaces.


[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

[Bal05] Baladi, Viviane Anisotropic Sobolev spaces and dynamical transfer operators: C foliations, Algebraic and topological dynamics (Contemporary Mathematics), Volume 385, American Mathematical Society, 2005, pp. 123-135 | DOI | MR | Zbl

[BJ20] Bonthonneau, Yannick Guedes; Jézéquel, Malo FBI Transform in Gevrey classes and Anosov flows (2020) (

[BKL02] Blank, Michael; Keller, Gerhard; Liverani, Carlangelo Ruelle–Perron–Frobenius spectrum for Anosov maps, Nonlinearity, Volume 15 (2002) no. 6, pp. 1905-1973 | DOI | MR | Zbl

[BL07] Butterley, Oliver; Liverani, Carlangelo Smooth Anosov flows: correlation spectra and stability, J. Mod. Dyn., Volume 1 (2007) no. 2, pp. 301-322 | DOI | MR | Zbl

[Bon20] Bonthonneau, Yannick Guedes Flow-independent Anisotropic space, and perturbation of resonances, Rev. Unión Mat. Argent., Volume 61 (2020) no. 1, pp. 63-72 | DOI | MR | Zbl

[BS12] Böttcher, Albrecht; Silbermann, Bernd Introduction to large truncated Toeplitz matrices, Universitext, Springer, 2012 | 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 | Zbl

[CDS01] Cannas Da Silva, Ana Lectures on Symplectic Geometry, Lecture Notes in Mathematics, 1764, Springer, 2001 | MR | Zbl

[DDZ12] Datchev, Kiril; Dyatlov, Semyon; Zworski, Maciej Sharp polynomial bounds on the number of Pollicott-Ruelle resonances, Ergodic Theory Dyn. Syst. (2012), pp. 1-16 | Zbl

[DFG15] Dyatlov, Semyon; Faure, Frédéric; Guillarmou, Colin Power spectrum of the geodesic flow on hyperbolic manifolds, Anal. PDE, Volume 8 (2015) no. 4, pp. 923-1000 | DOI | MR | Zbl

[DG16] Dyatlov, Semyon; Guillarmou, Colin Pollicott–Ruelle resonances for open systems, Ann. Henri Poincaré, Volume 17 (2016) no. 11, pp. 3089-3146 | DOI | MR | Zbl

[DR19] Dang, Nguyen Viet; Riviere, Gabriel Spectral analysis of Morse–Smale gradient flows, Ann. Sci. Éc. Norm. Supér., Volume 52 (2019) no. 6, pp. 1403-1458 | DOI | MR | Zbl

[Dya23] Dyatlov, Semyon Pollicott–Ruelle resolvent and Sobolev regularity, Pure Appl. Funct. Anal., Volume 8 (2023) no. 1, pp. 187-213 | 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

[EN99] Engel, Klaus-Jochen; Nagel, Rainer One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, 194, Springer, 1999 | Zbl

[Fal03] Falconer, Kenneth J. Fractal geometry: mathematical foundations and applications, John Wiley & Sons, 2003 | DOI | Zbl

[Fol89] Folland, Gerald B. Harmonic Analysis in phase space, Annals of Mathematics Studies, 122, Princeton University Press, 1989 | DOI | Zbl

[FR06] Faure, Frédéric; Roy, Nicolas Ruelle–Pollicott resonances for real analytic hyperbolic map, Nonlinearity, Volume 19 (2006) no. 6, pp. 1233-1252 | DOI | MR | Zbl

[FRS08] Faure, Frédéric; Roy, Nicolas; Sjöstrand, Johannes A semiclassical approach for Anosov Diffeomorphisms and Ruelle resonances, Open Math. J., Volume 1 (2008), pp. 35-81 | DOI | Zbl

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

[FT15] Faure, Frédéric; Tsujii, Masato Prequantum transfer operator for symplectic Anosov diffeomorphism, Astérisque, 375, Société Mathématique de France, 2015 | Zbl

[FT17] Faure, Frédéric; Tsujii, Masato The semiclassical zeta function for geodesic flows on negatively curved manifolds, Invent. Math., Volume 208 (2017) no. 3, pp. 851-998 | DOI | MR | Zbl

[FT21] Faure, Frédéric; Tsujii, Masato Microlocal analysis and Band structure of contact Anosov flows (2021) (

[GHW18] Guillarmou, Colin; Hilgert, Joachim; Weich, Tobias Classical and quantum resonances for hyperbolic surfaces, Math. Ann., Volume 370 (2018) no. 3-4, pp. 1231-1275 | DOI | MR | Zbl

[GL05] Gouëzel, Sébastien; Liverani, Carlangelo Banach spaces adapted to Anosov systems, Ergodic Theory Dyn. Syst., Volume 26 (2005) no. 1, pp. 189-217 | DOI | MR | Zbl

[GS94] Grigis, Alain; Sjöstrand, Johannes Microlocal analysis for differential operators. An introduction, London Mathematical Society Lecture Note Series, 196, Cambridge University Press, 1994 | DOI | Zbl

[Has94] Hasselblatt, Boris Regularity of the Anosov splitting and of horospheric foliations, Ergodic Theory Dyn. Syst., Volume 14 (1994) no. 4, pp. 645-666 | DOI | MR | Zbl

[HK90] Hurder, Steven E.; Katok, Anatole Differentiability, rigidity and Godbillon-Vey classes for Anosov flows, Publ. Math., Inst. Hautes Étud. Sci., Volume 72 (1990), pp. 5-61 | DOI | Numdam | Zbl

[HS86] Helffer, Bernard; Sjöstrand, Johannes Résonances en limite semi-classique. (Resonances in semi-classical limit), Mémoires de la Société Mathématique de France. Nouvelle Série, 24/25, Société Mathématique de France, 1986 | Zbl

[HS08] Hitrik, Michael; Sjöstrand, Johannes Rational invariant tori, phase space tunneling, and spectra for non-selfadjoint operators in dimension 2, Ann. Sci. Éc. Norm. Supér., Volume 41 (2008) no. 4, pp. 513-573 | DOI | Numdam | MR | Zbl

[Hör79] Hörmander, Lars The Weyl calculus of pseudo-differential operators, Commun. Pure Appl. Math., Volume 32 (1979) no. 3, pp. 359-443 | DOI | Zbl

[Hör83] Hörmander, Lars The analysis of linear partial differential operators. II: Differential operators with constant coefficients, Grundlehren der Mathematischen Wissenschaften, 257, Springer, 1983 | Zbl

[Hör03] Hörmander, Lars The Analysis of the Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis. Classics in Mathematics, Classics in Mathematics, Springer, 2003 (reprint of the 2nd edition 1990) | DOI | Zbl

[Jen18] Jenkinson, Oliver Ergodic optimization in dynamical systems, Ergodic Theory Dyn. Syst. (2018), pp. 1-26 | DOI | Zbl

[JZ17] Jin, Long; Zworski, Maciej A local trace formula for Anosov flows, Ann. Henri Poincaré, Volume 18 (2017) no. 1, pp. 1-35 | MR | Zbl

[Jéz20] Jézéquel, Malo Spectral theory for ultradifferentiable hyperbolic dynamics, Ph. D. Thesis, Sorbonne-Université, Paris, France (2020)

[Ler11] Lerner, Nicolas Metrics on the phase space and non-selfadjoint pseudo-differential operators, Pseudo-Differential Operators. Theory and Applications, 3, Springer, 2011 | Numdam | Zbl

[Mar02] Martinez, André An Introduction to Semiclassical and Microlocal Analysis, Universitext, Springer, 2002 | DOI | Zbl

[Med21] Meddane, Antoine A Morse complex for Axiom A flows (2021) ( | DOI

[MS98] McDuff, Dusa; Salamon, Dietmar Introduction to symplectic topology. 2nd edition, Oxford Mathematical Monographs, Clarendon Press, 1998 | Zbl

[NR11] Nicola, Fabio; Rodino, Luigi Global pseudo-differential calculus on Euclidean spaces, Pseudo-Differential Operators. Theory and Applications, 4, Springer, 2011 | Zbl

[NSZ14] Nonnenmacher, Stéphane; Sjöstrand, Johannes; Zworski, Maciej Fractal Weyl law for open quantum chaotic maps, Ann. Math., Volume 179 (2014) no. 1, pp. 179-251 | DOI | MR | Zbl

[Paz83] Pazy, André Semigroups of linear operators and applications to partial differential equations, 44, Springer, 1983 | DOI | Zbl

[Sjö90] Sjöstrand, Johannes Geometric bounds on the density of resonances for semiclassical problems, Duke Math. J., Volume 60 (1990) no. 1, pp. 1-57 | MR | Zbl

[Sjö96] Sjöstrand, Johannes Density of resonances for strictly convex analytic obstacles, Can. J. Math., Volume 48 (1996) no. 2, pp. 397-447 (with an appendix by M. Zworski) | DOI | MR | Zbl

[Sjö00] Sjöstrand, Johannes Asymptotic distribution of eigenfrequencies for damped wave equations, Publ. Res. Inst. Math. Sci., Volume 36 (2000) no. 5, pp. 573-611 | DOI | Numdam | MR | Zbl

[Tay96a] Taylor, Michael E. Partial differential equations. Vol. I: Basic theory, Applied Mathematical Sciences, 115, Springer, 1996 | Zbl

[Tay96b] Taylor, Michael E. Partial differential equations. Vol. II: Qualitative studies of linear equations, Applied Mathematical Sciences, 116, Springer, 1996 | DOI | Zbl

[TE05] Trefethen, Lloyd N.; Embree, Mark Spectra and pseudospectra. The behavior of nonnormal matrices and operators, Princeton University Press, 2005 | DOI | Zbl

[Tsu10] Tsujii, Masato Quasi-compactness of transfer operators for contact Anosov flows, Nonlinearity, Volume 23 (2010) no. 7, pp. 1495-1545 | DOI | MR | Zbl

[Tsu12] Tsujii, Masato Contact Anosov flows and the Fourier–Bros–Iagolnitzer transform, Ergodic Theory Dyn. Syst., Volume 32 (2012) no. 6, pp. 2083-2118 | DOI | MR | Zbl

[WZ01] Wunsch, Jared; Zworski, Maciej The FBI transform on compact 𝒞 manifolds., Trans. Am. Math. Soc., Volume 353 (2001) no. 3, pp. 1151-1167 | DOI | MR | Zbl

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