The Persistence of ω-Limit Sets under Perturbation of the Generating Function

Abstract

We consider the set valued function $\Omega $ taking $f$ in $\ C(I,I)$ to its collection of $\omega $-limit sets $\Omega (f)=\{\omega (x,f):x\in I\}$, and consider how $\Omega (f)$ is affected by pertubations of $f$. Our main result characterizes those functions $f$ in $C(I,I)$ at which $\Omega $ is upper semicontinuous, so that whenever $g$ is sufficiently close to $f$, every $\omega $-limit set of $g$ is close to some $\omega $-limit set of $f$ in the Hausdorff metric space. We also develop necessary and sufficient conditions for a function $f$ in $C(I,I)$ to be a point of lower semicontinuity of the map $\Omega$.