# A Conley-type decomposition of the strong chain recurrent set

## TL;DR

This paper establishes a Conley-type decomposition for the strong chain recurrent set of a continuous flow on a compact metric space, introducing strongly stable sets that serve as attractor-like components in the decomposition.

## Contribution

It introduces the concept of strongly stable sets and proves they form the basis for a Conley-type decomposition of the strong chain recurrent set.

## Key findings

- Strong chain recurrent set can be decomposed using strongly stable sets.
- Strongly stable sets are invariant and contain attractors.
- The strong chain recurrent set equals the intersection of all strongly stable sets and their complements.

## Abstract

For a continuous flow on a compact metric space, the aim of this paper is to prove a Conley-type decomposition of the strong chain recurrent set. We first discuss in details the main properties of strong chain recurrent sets. We then introduce the notion of strongly stable set as an invariant set which is the intersection of the $\omega$-limits of a specific family of nested and definitively invariant neighborhoods of itself. This notion strengthens the one of stable set; moreover, any attractor results strongly stable. We then show that strongly stable sets play the role of attractors in the decomposition of the strong chain recurrent set; indeed, we prove that the strong chain recurrent set coincides with the intersection of all strongly stable sets and their complementaries.