# Spectral analysis of Morse-Smale flows I: construction of the anisotropic spaces

## TL;DR

This paper establishes the existence of a discrete correlation spectrum for Morse-Smale flows on smooth forms by constructing anisotropic Sobolev spaces where the Lie derivative exhibits a discrete spectrum, advancing spectral analysis in dynamical systems.

## Contribution

It introduces a novel construction of anisotropic Sobolev spaces tailored for Morse-Smale flows, enabling spectral analysis of the Lie derivative on smooth forms.

## Key findings

- Discrete correlation spectrum exists for Morse-Smale flows.
- Constructed anisotropic Sobolev spaces with desired spectral properties.
- Lie derivative has a discrete spectrum on these spaces.

## Abstract

We prove the existence of a discrete correlation spectrum for Morse-Smale flows acting on smooth forms on a compact manifold. This is done by constructing spaces of currents with anisotropic Sobolev regularity on which the Lie derivative has a discrete spectrum.