Spectral analysis of Morse-Smale gradient flows
Document type :
Compte-rendu et recension critique d'ouvrage
DOI :
Title :
Spectral analysis of Morse-Smale gradient flows
Author(s) :
Dang, Nguyen Viet [Auteur]
Institut Camille Jordan [ICJ]
Probabilités, statistique, physique mathématique [PSPM]
Riviere, Gabriel [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Institut Camille Jordan [ICJ]
Probabilités, statistique, physique mathématique [PSPM]
Riviere, Gabriel [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Journal title :
Annales Scientifiques de l'École Normale Supérieure
Publisher :
Société mathématique de France
Publication date :
2019
ISSN :
0012-9593
English keyword(s) :
Morse theory
gradient flow
resonances
transfer operator
differential topology
gradient flow
resonances
transfer operator
differential topology
HAL domain(s) :
Mathématiques [math]/Systèmes dynamiques [math.DS]
Mathématiques [math]/Topologie géométrique [math.GT]
Mathématiques [math]/Théorie spectrale [math.SP]
Mathématiques [math]/Topologie géométrique [math.GT]
Mathématiques [math]/Théorie spectrale [math.SP]
English abstract : [en]
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 ...
Show more >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.Show less >
Show more >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.Show less >
Language :
Anglais
Popular science :
Non
Comment :
Shortened version (56 p.), to appear in Annales Sci. ENS
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Source :
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