A Morse complex for Axiom A flows

Antoine Meddane (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

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 verifying Smale's transversality assumptions. 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.

Mots clés: Axiom A flows; spectral theory; Morse complex; microlocal analysis; Pollicott-Ruelle resonances

Domaines: Systèmes dynamiques [math.DS]; Théorie spectrale [math.SP]; Topologie géométrique [math.GT]

Antoine Meddane. A Morse complex for Axiom A flows. 2021. ⟨hal-03287355⟩