Topological Sensitivity on Hyperspaces

Abstract

We introduce and study topological asymptotic sensitivity, Li-Yorke topological sensitivity, pointwise topological sensitivity for general topological spaces. We study relations among these variants of topological sensitivities and $\mathcal F$ topological sensitivity, strong topological sensitivity and multi topological sensitivity. We prove that strong topological sensitivity (respectively multi topological sensitivity) on the hyperspace dynamical system is equivalent to strong topological sensitivity (respectively multi topological sensitivity) of a dynamical system for uniform compact spaces. It is shown that Li-Yorke sensitivity and Li-Yorke topological sensitivity are equivalent on compact metric spaces. We also show that on compact Hausdorff topological spaces topological sensitivity implies topological asymptotic sensitivity. Moreover, we provide necessary examples and counterexamples.