Persistence in Infinite-Dimensional Systems

Abstract

Summary:In this paper, we discuss the properties of limit sets of subsets and attractors in a compact metric space. It is shown that the $\omega $-limit set $\omega (Y)$ of $Y$ is the limit point of the sequence $\lbrace (\mathop {\mathrm Cl}Y)\cdot [i,\infty )\rbrace _{i=1}^{\infty }$ in $2^X$ and also a quasi-attractor is the limit point of attractors with respect to the Hausdorff metric. It is shown that if a component of an attractor is not an attractor, then it must be a real quasi-attractor