A Morse complex for Axiom A flows

TL;DR

This paper develops a finite-dimensional cohomological complex for Axiom A flows on compact manifolds, linking it to De Rham cohomology and generalizing Morse complexes through anisotropic Sobolev spaces.

Contribution

It introduces a novel cohomological complex invariant under Axiom A flows, extending Morse theory to more general dynamical systems using anisotropic Sobolev spaces.

Findings

The complex's cohomology matches De Rham cohomology.

In Morse-Smale cases, it reduces to the classical Morse complex.

Provides a new analytical framework for studying Axiom A flows.

Abstract

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.