[Analyse spectrale des flots de gradients Morse-Smale]
Sur une variété lisse, compacte et orientée sans bord, nous donnons une description complète de la fonction de corrélation des flots de gradients Morse-Smale vérifiant certaines hypothèses de non-résonance. Ce résultat est obtenu en analysant précisément le spectre du générateur d'un tel flot agissant sur certains espaces de Sobolev anisotropes. Nous démontrons en particulier que ce spectre dynamique est donné par des combinaisons linéaires à coefficients entiers des exposants de Lyapunov aux points critiques de la fonction de Morse. Grâce à cette analyse spectrale et en analogie complète avec la théorie de Hodge-de Rham, nous interprétons le complexe de Morse comme l'image du complexe de de Rham par le projecteur sur le noyau du générateur du flot. Ceci nous permet de retrouver des résultats classiques de topologie différentielle comme les inégalités de Morse et la dualité de Poincaré.
On a smooth, compact and oriented manifold without boundary, we give a complete description of the correlation function of a Morse-Smale gradient flow satisfying a certain nonresonance assumption. This is done by analyzing precisely the spectrum of the generator of such a flow acting on certain anisotropic spaces of currents. In particular, we prove that this dynamical spectrum is given by linear combinations with integer coefficients of the Lyapunov exponents at the critical points of the Morse function. Via this spectral analysis and in analogy with Hodge-de Rham theory, we give an interpretation of the Morse complex as the image of the de Rham complex under the spectral projector on the kernel of the generator of the flow. This allows us to recover classical results from differential topology such as the Morse inequalities and Poincaré duality.
DOI : 10.24033/asens.2412
Keywords: Hyperbolic dynamical systems, Morse-Smale flows, Pollicott-Ruelle resonances, anisotropic Sobolev spaces, microlocal analysis, Morse complex.
Mot clés : Systèmes dynamiques hyperboliques, flots Morse-Smale, résonances de Pollicott-Ruelle, espaces de Sobolev anisotropes, analyse microlocale, complexe de Morse.
@article{ASENS_2019__52_6_1403_0,
author = {Dang, Nguyen Viet and Rivi\`ere, Gabriel},
title = {Spectral analysis of {Morse-Smale} gradient flows},
journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
pages = {1403--1458},
publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
volume = {Ser. 4, 52},
number = {6},
year = {2019},
doi = {10.24033/asens.2412},
mrnumber = {4061023},
zbl = {1448.37029},
language = {en},
url = {http://www.numdam.org/articles/10.24033/asens.2412/}
}
TY - JOUR AU - Dang, Nguyen Viet AU - Rivière, Gabriel TI - Spectral analysis of Morse-Smale gradient flows JO - Annales scientifiques de l'École Normale Supérieure PY - 2019 SP - 1403 EP - 1458 VL - 52 IS - 6 PB - Société Mathématique de France. Tous droits réservés UR - http://www.numdam.org/articles/10.24033/asens.2412/ DO - 10.24033/asens.2412 LA - en ID - ASENS_2019__52_6_1403_0 ER -
%0 Journal Article %A Dang, Nguyen Viet %A Rivière, Gabriel %T Spectral analysis of Morse-Smale gradient flows %J Annales scientifiques de l'École Normale Supérieure %D 2019 %P 1403-1458 %V 52 %N 6 %I Société Mathématique de France. Tous droits réservés %U http://www.numdam.org/articles/10.24033/asens.2412/ %R 10.24033/asens.2412 %G en %F ASENS_2019__52_6_1403_0
Dang, Nguyen Viet; Rivière, Gabriel. Spectral analysis of Morse-Smale gradient flows. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 6, pp. 1403-1458. doi : 10.24033/asens.2412. http://www.numdam.org/articles/10.24033/asens.2412/
A Lefschetz fixed point formula for elliptic complexes. I, Ann. of Math., Volume 86 (1967), pp. 374-407 (ISSN: 0003-486X) | DOI | MR | Zbl
Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov., Volume 90 (1967), 209 pages (ISSN: 0371-9685) | MR | Zbl
Continuity of the fundamental operations on distributions having a specified wave front set (with a counterexample by Semyon Alesker), Studia Math., Volume 232 (2016), pp. 201-226 (ISSN: 0039-3223) | MR | Zbl
Spectral structure of transfer operators for expanding circle maps, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 34 (2017), pp. 31-43 (ISSN: 0294-1449) | DOI | Numdam | MR | Zbl
Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity, Volume 15 (2002), pp. 1905-1973 (ISSN: 0951-7715) | DOI | MR | Zbl
Smooth Anosov flows: correlation spectra and stability, J. Mod. Dyn., Volume 1 (2007), pp. 301-322 (ISSN: 1930-5311) | DOI | MR | Zbl
Lower bounds for the Ruelle spectrum of analytic expanding circle maps, Ergodic Theory Dynam. Systems, Volume 39 (2019), pp. 289-310 (ISSN: 0143-3857) | DOI | MR | Zbl
Morse theory indomitable, Inst. Hautes Études Sci. Publ. Math., Volume 68 (1988), pp. 99-114 (ISSN: 0073-8301) | DOI | Numdam | MR | Zbl
The ergodic theory of Axiom A flows, Invent. math., Volume 29 (1975), pp. 181-202 (ISSN: 0020-9910) | DOI | MR | Zbl
Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms, Ann. Inst. Fourier, Volume 57 (2007), pp. 127-154 http://aif.cedram.org/item?id=AIF_2007__57_1_127_0 (ISSN: 0373-0956) | DOI | Numdam | MR | Zbl
, Geometric and probabilistic structures in dynamics (Contemp. Math.), Volume 469, Amer. Math. Soc., 2008, pp. 29-68 | DOI | MR | Zbl
Equivalence and decomposition of vector fields about an elementary critical point, Amer. J. Math., Volume 85 (1963), pp. 693-722 (ISSN: 0002-9327) | DOI | MR | Zbl
The extension of distributions on manifolds, a microlocal approach, Ann. Henri Poincaré, Volume 17 (2016), pp. 819-859 (ISSN: 1424-0637) | DOI | MR | Zbl
, Grundl. math. Wiss., 266, Springer, 1984, 167 pages (ISBN: 3-540-13463-8) | DOI | MR | Zbl
Pollicott-Ruelle resonances for open systems, Ann. Henri Poincaré, Volume 17 (2016), pp. 3089-3146 (ISSN: 1424-0637) | DOI | MR | Zbl
On decay of correlations in Anosov flows, Ann. of Math., Volume 147 (1998), pp. 357-390 (ISSN: 0003-486X) | DOI | MR | Zbl
Equidistribution of the conormal cycle of random nodal sets, J. Eur. Math. Soc. (JEMS), Volume 20 (2018), pp. 3017-3071 (ISSN: 1435-9855) | DOI | MR | Zbl
Dynamical zeta functions for Anosov flows via microlocal analysis, Ann. Sci. Éc. Norm. Supér., Volume 49 (2016), pp. 543-577 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl
, Graduate Studies in Math., 200, 2019 | MR
, Graduate Texts in Math., 194, Springer, 2000, 586 pages (ISBN: 0-387-98463-1) | MR | Zbl
Instantons beyond topological theory. I, J. Inst. Math. Jussieu, Volume 10 (2011), pp. 463-565 (ISSN: 1474-7480) | DOI | MR | Zbl
Semi-classical approach for Anosov diffeomorphisms and Ruelle resonances, Open Math. J., Volume 1 (2008), pp. 35-81 | DOI | MR | Zbl
Upper bound on the density of Ruelle resonances for Anosov flows, Comm. Math. Phys., Volume 308 (2011), pp. 325-364 (ISSN: 0010-3616) | DOI | MR | Zbl
Astérisque, Astérisque, 375, 2015, 222 pages (ISBN: 978-2-85629-823-7, ISSN: 0303-1179) | MR
The semiclassical zeta function for geodesic flows on negatively curved manifolds, Inventiones math., Volume 208 (2017), pp. 851-998 | DOI | MR | Zbl
Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties, J. Differential Geom., Volume 79 (2008), pp. 433-477 http://projecteuclid.org/euclid.jdg/1213798184 (ISSN: 0022-040X) | MR | Zbl
Anosov flows and dynamical zeta functions, Ann. of Math., Volume 178 (2013), pp. 687-773 (ISSN: 0003-486X) | DOI | MR | Zbl
Spectre du flot géodésique en courbure négative [d'après F. Faure et M. Tsujii], Séminaire Bourbaki, Vol. 2014/2015, exp. no 1098, Astérisque, Volume 380 (2016), pp. 325-353 (ISBN: 978-2-85629-836-7, ISSN: 0303-1179) | MR | Zbl
A lemma in the theory of structural stability of differential equations, Proc. Amer. Math. Soc., Volume 11 (1960), pp. 610-620 (ISSN: 0002-9939) | DOI | MR | Zbl
, Surveys in differential geometry (Surv. Differ. Geom.), Volume 7, Int. Press, 2000, pp. 259-311 | DOI | MR | Zbl
Finite volume flows and Morse theory, Ann. of Math., Volume 153 (2001), pp. 1-25 (ISSN: 0003-486X) | DOI | MR | Zbl
, Springer Study Edition, Springer, 1990, 440 pages (ISBN: 3-540-52343-X) | DOI | MR | Zbl
Résonances en limite semi-classique, Mém. Soc. Math. France (N.S.), Volume 24-25 (1986), pp. 1-228 (ISSN: 0037-9484) | Numdam | MR | Zbl
, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 59, Tata Institute of Fundamental Research, Bombay; by the Narosa Publishing House, New Delhi, 1978, 175 pages (ISBN: 0-387-08944-6) | MR | Zbl
, Éditions de l'École polytechnique, Palaiseau, 2012, 182 pages (ISBN: 978-2-7302-1585-5) | MR
On the Thom-Smale complex, An extension of a theorem by Cheeger and Müller (Bismut, J.-M.; Zhang, W., eds.) (Astérisque), Volume 205 (1992) (ISSN: 0303-1179) | MR
On contact Anosov flows, Ann. of Math., Volume 159 (2004), pp. 1275-1312 (ISSN: 0003-486X) | DOI | MR | Zbl
A current approach to Morse and Novikov theories, Rend. Mat. Appl., Volume 36 (2015), pp. 95-195 (ISSN: 1120-7183) | MR | Zbl
, Mathematical Notes, Princeton Univ. Press, N.J.; University of Tokyo Press, Tokyo, 1969, 118 pages | MR | Zbl
Memoirs of the American Mathematical Society, Mem. Amer. Math. Soc., 208, 2010 (ISBN: 978-0-8218-4870-8, ISSN: 0065-9266) | DOI | MR | Zbl
On the rate of mixing of Axiom A flows, Invent. math., Volume 81 (1985), pp. 413-426 (ISSN: 0020-9910) | DOI | MR | Zbl
One-dimensional Gibbs states and Axiom A diffeomorphisms, J. Differential Geom., Volume 25 (1987), pp. 117-137 http://projecteuclid.org/euclid.jdg/1214440727 (ISSN: 0022-040X) | MR | Zbl
Resonances for Axiom flows, J. Differential Geom., Volume 25 (1987), pp. 99-116 http://projecteuclid.org/euclid.jdg/1214440726 (ISSN: 0022-040X) | MR | Zbl
, Hermann, 1966, 420 pages | MR | Zbl
Gibbs measures in ergodic theory, Uspehi Mat. Nauk, Volume 27 (1972), pp. 21-64 (ISSN: 0042-1316) | MR | Zbl
Morse inequalities for a dynamical system, Bull. Amer. Math. Soc., Volume 66 (1960), pp. 43-49 (ISSN: 0002-9904) | DOI | MR | Zbl
On gradient dynamical systems, Ann. of Math., Volume 74 (1961), pp. 199-206 (ISSN: 0003-486X) | DOI | MR | Zbl
Differentiable dynamical systems, Bull. Amer. Math. Soc., Volume 73 (1967), pp. 747-817 (ISSN: 0002-9904) | DOI | MR | Zbl
Sur une partition en cellules associée à une fonction sur une variété, C. R. Acad. Sci. Paris, Volume 228 (1949), pp. 973-975 (ISSN: 0001-4036) | MR | Zbl
Quasi-compactness of transfer operators for contact Anosov flows, Nonlinearity, Volume 23 (2010), pp. 1495-1545 (ISSN: 0951-7715) | DOI | MR | Zbl
Contact Anosov flows and the Fourier-Bros-Iagolnitzer transform, Ergodic Theory Dynam. Systems, Volume 32 (2012), pp. 2083-2118 (ISSN: 0143-3857) | DOI | MR | Zbl
The Morse-Witten complex via dynamical systems, Expo. Math., Volume 24 (2006), pp. 127-159 (ISSN: 0723-0869) | DOI | MR | Zbl
Supersymmetry and Morse theory, J. Differential Geom., Volume 17 (1982), pp. 661-692 http://projecteuclid.org/euclid.jdg/1214437492 (ISSN: 0022-040X) | MR | Zbl
, Graduate Studies in Math., 138, Amer. Math. Soc., 2012, 431 pages (ISBN: 978-0-8218-8320-4) | DOI | MR | Zbl
Cité par Sources :
