Spectral analysis of morse-smale gradient flows

TL;DR

This paper provides a spectral analysis of Morse-Smale gradient flows on manifolds, linking the spectrum to Lyapunov exponents and classical topological results like Morse inequalities.

Contribution

It offers a complete spectral description of the generator of Morse-Smale flows and connects it to Morse theory and classical topology.

Findings

Spectrum given by integer combinations of Lyapunov exponents

Spectral projector recovers Morse complex from de Rham complex

Classical topological results derived from spectral analysis

Abstract

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 Poincar{\'e} duality.