Minimal and ω-Minimal Sets of Functions with Connected Gδ Graphs

Abstract

Let $I=[0,1]$, and let $\mathcal J$ be the class of functions $I \rightarrow I$ with connected $G_{\delta}$ graph. Recently it was shown that dynamical systems generated by maps in $\mathcal J$ have some nice properties. Thus, the Sharkovsky's theorem is true, and a map has zero topological entropy if and only if every periodic point has period $2^{n}$, for an integer $n\ge 0$. In this paper we consider, for a map $\varphi$ in $\mathcal J$, properties of $\omega$-minimal sets; i.e., sets $M\subset I$ such that the $\omega$-limit set $\omega_{\varphi}(x)$ is $M$, for every $x \in M$. If $\varphi $ is continuous, then, as is well-known, $M$ is minimal if and only if $M$ is non-empty, closed, $\varphi(M)\subseteq M$, any point in $M$ is uniformly recurrent, and no proper subset of $M$ has this property. In this paper we prove that the same is true for $\varphi\in\mathcal J$ with zero topological entropy, but not for an arbitrary $\varphi\in\mathcal J$.