A Conley-type decomposition of the strong chain recurrent set

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 detail the main properties of strong chain recurrent sets. We then introduce the notion of strongly stable set as a closed 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 that of stable set; moreover, any attractor turns out to be 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 complementary.