[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://archive.numdam.org/articles/10.24033/asens.2412/}
}
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%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://archive.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://archive.numdam.org/articles/10.24033/asens.2412/
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