Lusternik-Schnirelmann Theory for a Morse Decomposition

Abstract

Let ${\varphi }^t$ 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 $\{M_i{\}}_{i=1}^n$. For every category $\nu$ on $N/L$, we prove that $\nu (N/L)\le \nu ([L])+{\sum }_{i=1}^n\nu (M_i)$. As a result if ${\varphi }^t{|}_I$ is gradient-like and $X$ is semi-locally contractible, then ${\varphi }^t$ 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.

9 pages