A Morse complex for Axiom A flows

Antoine Meddane

doi:10.5802/jep.299

A Morse complex for Axiom A flows
[Un complexe de Morse pour les flots Axiome A]

Antoine Meddane ¹

¹ Laboratoire de mathématiques Jean Leray (UMR CNRS 6629), Université de Nantes, 2 rue de la Houssinière, BP92208, 44322 Nantes Cédex 3, France

Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 641-712.

Abstract (VO)
Résumé

On a smooth compact Riemannian manifold without boundary, we construct a finite dimensional cohomological complex of currents that are invariant by an Axiom A flow satisfying the strong transversality assumption. The cohomology of that complex is isomorphic to the de Rham cohomology via certain spectral projectors. This construction is achieved by defining anisotropic Sobolev spaces adapted to the global dynamics of Axiom A flows. In the particular case of Morse-Smale gradient flows, this complex coincides with the classical Morse complex.

Sur une variété riemannienne compacte, lisse et sans bord, nous construisons un complexe cohomologique de courants, de dimension finie, invariant par un flot Axiome A vérifiant l’hypothèse de forte transversalité. La cohomologie de ce complexe est isomorphe à la cohomologie de de Rham via certains projecteurs spectraux. Cette construction est réalisée en définissant des espaces de Sobolev anisotropes adaptés à la dynamique globale des flots Axiome A. Dans le cas particulier des flots de gradient de Morse-Smale, ce complexe coïncide avec le complexe de Morse classique.

Reçu le : 2021-09-16
Accepté le : 2025-05-02
Publié le : 2025-05-15

DOI : 10.5802/jep.299

Classification : 58J50
Keywords: Hyperbolic dynamical systems, Axiom A flows, Pollicott-Ruelle resonances, Morse complex, anisotropic Sobolev spaces, microlocal analysis
Mots-clés : Systèmes dynamiques hyperboliques, flots Axiome A, résonances de Pollicott-Ruelle, complexe de Morse, espaces de Sobolev anisotropes, analyse microlocale

Affiliations des auteurs :

Antoine Meddane ¹

¹ Laboratoire de mathématiques Jean Leray (UMR CNRS 6629), Université de Nantes, 2 rue de la Houssinière, BP92208, 44322 Nantes Cédex 3, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Antoine Meddane. A Morse complex for Axiom A flows. Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 641-712. doi : 10.5802/jep.299. https://jep.centre-mersenne.org/articles/10.5802/jep.299/

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