Ruelle spectrum of linear pseudo-Anosov maps
Journal de l’École polytechnique - Mathématiques, Volume 6 (2019), pp. 811-877.

The Ruelle resonances of a dynamical system are spectral data describing the precise asymptotics of correlations. We classify them completely for a class of chaotic two-dimensional maps, the linear pseudo-Anosov maps, in terms of the action of the map on cohomology. As applications, we obtain a full description of the distributions which are invariant under the linear flow in the stable direction of such a linear pseudo-Anosov map, and we solve the cohomological equation for this flow.

Les résonances de Ruelle d’un système dynamique sont des données spectrales qui décrivent les asymptotiques précises des corrélations. Nous les classifions complètement pour une classe d’applications chaotiques en dimension deux, les applications pseudo-Anosov linéaires, en termes de l’action en cohomologie de la transformation. Nous en déduisons une description complète des distributions qui sont invariantes par le flot linéaire dans la direction stable d’un tel pseudo-Anosov, et nous résolvons l’équation cohomologique pour ce flot.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.107
Classification: 37D50,  37A25
Keywords: Ruelle resonances, pseudo-Anosov, linear flow; cohomological equation
Faure, Frédéric 1; Gouëzel, Sébastien 2; Lanneau, Erwan 1

1 Univ. Grenoble Alpes, CNRS UMR 5582, Institut Fourier F-38000 Grenoble, France
2 Laboratoire Jean Leray, CNRS UMR 6629, Université de Nantes 2 rue de la Houssinière, 44322 Nantes, France
@article{JEP_2019__6__811_0,
     author = {Faure, Fr\'ed\'eric and Gou\"ezel, S\'ebastien and Lanneau, Erwan},
     title = {Ruelle spectrum of linear {pseudo-Anosov} maps},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique - Math\'ematiques},
     pages = {811--877},
     publisher = {Ecole polytechnique},
     volume = {6},
     year = {2019},
     doi = {10.5802/jep.107},
     zbl = {07114039},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jep.107/}
}
TY  - JOUR
AU  - Faure, Frédéric
AU  - Gouëzel, Sébastien
AU  - Lanneau, Erwan
TI  - Ruelle spectrum of linear pseudo-Anosov maps
JO  - Journal de l’École polytechnique - Mathématiques
PY  - 2019
DA  - 2019///
SP  - 811
EP  - 877
VL  - 6
PB  - Ecole polytechnique
UR  - http://www.numdam.org/articles/10.5802/jep.107/
UR  - https://zbmath.org/?q=an%3A07114039
UR  - https://doi.org/10.5802/jep.107
DO  - 10.5802/jep.107
LA  - en
ID  - JEP_2019__6__811_0
ER  - 
%0 Journal Article
%A Faure, Frédéric
%A Gouëzel, Sébastien
%A Lanneau, Erwan
%T Ruelle spectrum of linear pseudo-Anosov maps
%J Journal de l’École polytechnique - Mathématiques
%D 2019
%P 811-877
%V 6
%I Ecole polytechnique
%U https://doi.org/10.5802/jep.107
%R 10.5802/jep.107
%G en
%F JEP_2019__6__811_0
Faure, Frédéric; Gouëzel, Sébastien; Lanneau, Erwan. Ruelle spectrum of linear pseudo-Anosov maps. Journal de l’École polytechnique - Mathématiques, Volume 6 (2019), pp. 811-877. doi : 10.5802/jep.107. http://www.numdam.org/articles/10.5802/jep.107/

[Ada17] Adam, Alexander Generic non-trivial resonances for Anosov diffeomorphisms, Nonlinearity, Volume 30 (2017) no. 3, pp. 1146-1164 | DOI | MR | Zbl

[AG13] Avila, Artur; Gouëzel, Sébastien Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow, Ann. of Math. (2), Volume 178 (2013) no. 2, pp. 385-442 | DOI | Zbl

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

[Bal17] Baladi, Viviane The quest for the ultimate anisotropic Banach space, J. Statist. Phys., Volume 166 (2017) no. 3-4, pp. 525-557 | DOI | MR | Zbl

[BJ08] Bandtlow, Oscar F.; Jenkinson, Oliver Explicit eigenvalue estimates for transfer operators acting on spaces of holomorphic functions, Adv. Math., Volume 218 (2008) no. 3, pp. 902-925 | DOI | MR | Zbl

[BJS13] Bandtlow, Oscar F.; Just, Wolfram; Slipantschuk, Julia Analytic expanding circle maps with explicit spectra, Nonlinearity, Volume 26 (2013) no. 12, pp. 3231-3245 | DOI | MR | Zbl

[BJS17] Bandtlow, Oscar F.; Just, Wolfram; Slipantschuk, Julia Complete spectral data for analytic Anosov maps of the torus, Nonlinearity, Volume 30 (2017) no. 7, pp. 2667-2686 | DOI | MR

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

[BT08] Baladi, Viviane; Tsujii, Masato Dynamical determinants and spectrum for hyperbolic diffeomorphisms, Geometric and probabilistic structures in dynamics (Contemp. Math.), Volume 469, American Mathematical Society, Providence, RI, 2008, pp. 29-68 | DOI | MR | Zbl

[Buf14a] Bufetov, Alexander I. Finitely-additive measures on the asymptotic foliations of a Markov compactum, Moscow Math. J., Volume 14 (2014) no. 2, pp. 205-224 | DOI | MR | Zbl

[Buf14b] Bufetov, Alexander I. Limit theorems for translation flows, Ann. of Math. (2), Volume 179 (2014) no. 2, pp. 431-499 | DOI | MR | 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

[For97] Forni, Giovanni Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus, Ann. of Math. (2), Volume 146 (1997) no. 2, pp. 295-344 | DOI | MR | Zbl

[For02] Forni, Giovanni Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), Volume 155 (2002) no. 1, pp. 1-103 | DOI | MR | Zbl

[For07] Forni, Giovanni Sobolev regularity of solutions of the cohomological equation, 2007 | arXiv

[For18] Forni, Giovanni Ruelle resonances and cohomological equations (2018) (Talk at the Teichmüller dynamics, mapping class groups and applications summer school in CIRM)

[GL08] Gouëzel, Sébastien; Liverani, Carlangelo Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties, J. Differential Geom., Volume 79 (2008) no. 3, pp. 433-477 | DOI | MR | Zbl

[GL19] Giulietti, Paolo; Liverani, Carlangelo Parabolic dynamics and anisotropic Banach spaces, J. Eur. Math. Soc. (JEMS), Volume 21 (2019) no. 9, pp. 2793-2858 | DOI | MR

[Hen93] Hennion, Hubert Sur un théorème spectral et son application aux noyaux lipchitziens, Proc. Amer. Math. Soc., Volume 118 (1993) no. 2, pp. 627-634 | MR | Zbl

[HK95] Hasselblatt, Boris; Katok, Anatole Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995 | MR | Zbl

[Hör03] Hörmander, Lars The analysis of linear partial differential operators. I, Classics in Mathematics, Springer-Verlag, Berlin, 2003 | DOI | Zbl

[Jéz17] Jézéquel, Malo Local and global trace formulae for smooth hyperbolic diffeomorphisms, 2017 | arXiv

[MMY05] Marmi, Stefano; Moussa, Pierre; Yoccoz, Jean-Christophe The cohomological equation for Roth-type interval exchange maps, J. Amer. Math. Soc., Volume 18 (2005) no. 4, pp. 823-872 | DOI | MR

[MY16] Marmi, Stefano; Yoccoz, Jean-Christophe Hölder regularity of the solutions of the cohomological equation for Roth type interval exchange maps, Comm. Math. Phys., Volume 344 (2016) no. 1, pp. 117-139 | DOI | Zbl

[Nau12] Naud, Frédéric The Ruelle spectrum of generic transfer operators, Discrete Contin. Dynam. Systems, Volume 32 (2012) no. 7, pp. 2521-2531 | DOI | MR | Zbl

[Rue90] Ruelle, David An extension of the theory of Fredholm determinants, Publ. Math. Inst. Hautes Études Sci., Volume 72 (1990), pp. 175-193 | DOI | Zbl

[Thu88] Thurston, William P. On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.), Volume 19 (1988) no. 2, pp. 417-431 | DOI | MR | Zbl

[Zor06] Zorich, Anton Flat surfaces, Frontiers in number theory, physics, and geometry. I, Springer, Berlin, 2006, pp. 437-583 | DOI | Zbl

Cited by Sources: