The Set of Continuous Functions with Zero Topological Entropy

Abstract

Let $I=[0,1]$. We show that those functions in $C(I,I)$ possessing zero topological entropy form a nowhere dense perfect subset of the continuous self maps of the interval. We also show that every function with zero topological entropy that possesses an infinite $\omega $-limit set is the uniform limit of functions having only finite $\omega $-limit sets.