Lusternik-Schnirelmann Theory for a Morse Decomposition

TL;DR

This paper extends Lusternik-Schnirelmann theory to Morse decompositions of continuous flows, providing bounds on categories and implications for the existence of rest points in gradient-like systems.

Contribution

It establishes a new inequality relating categories of invariant sets and Morse decompositions, and applies it to guarantee rest points in gradient-like flows on semi-locally contractible spaces.

Findings

Proves a category inequality for Morse decompositions.

Shows existence of rest points based on Conley index and category.

Extends Lusternik-Schnirelmann theory to Morse decompositions.

Abstract

Let $\phi^t$ be a continuous flow on a metric space $X$ and $I$ be an isolated invariant set with an index pair $(N,L)$ and a Morse decomposition $\{M_i\}^n_{i=1}$. For every category $\nu$ on $N/L$, we prove that $\nu(N/L)\leq \nu([L])+\sum_{i=1}^n \nu(M_i)$. As a result if $\phi^t|_I$ is gradient-like and $X$ is semi-locally contractible, then $\phi^t$ has at least $\nu_H(h(I))-1$ rest points in $I$ where $h(I)$ is the Conley index of $I$ and $\nu_H$ is the Homotopy Lusternik-Schnirelmann category.