Chaos and the recurrent set.

Abstract

Let $f$ be an element of $C(I,I)$ with $R(f)=\{x\in I:x\in \omega (x,f)\}$ its recurrent set. We study the relationship between the structure of $R(f)$ and the chaotic nature of the function $f$ . We show that $R(f)$ is always a $G_{\delta }$ set whenever $f$ has zero topological entropy, although $R(f)$ is closed for the typical continuous function $f$ with zero topological entropy. We also develop necessary and sufficient conditions on $f$ for $R(f)$ to be closed.