Abstract topological dynamics involving set-valued functions (Q2187145)

The main results of the paper can be summarized as: \((1)\) Let \(X\) be an infinite set and \(T : X \multimap X\) any set-valued map with nonempty values. Then there exists a compact \(T_1\) topology on \(X\) with respect to which \(T\) is lower semicontinuous; \((2)\) If \(X\) is an infinite compact Hausdorff space and \(T : X \multimap X\) is a closed-valued upper semicontinuous set-valued map then \[ T\bigl(\bigcap_{n \in \mathbb{N}}T^n(X)\bigr) = \bigcap_{n \in \mathbb{N}}T^n(X) \neq \emptyset. \] Some illustrative examples are given.