Continuity Structure of f→∪x∈Iω(x,f) and f→{ω,f:x∈I}

Abstract

Let the maps $\Lambda $ and $\Omega $ be defined on $C(I,I)$ so that $ f\longmapsto \Lambda (f)=\cup _{x\in I}\omega (x,f)$ and $f\longmapsto \Omega (f)=\{\omega (x,f):x\in I\}.$ We characterize those functions at which $\Lambda $ is continuous, as well as those functions at which $\Omega $ is continuous when its domain is restricted to those elements of $C(I,I)$ possessing zero topological entropy.