A fixed point theorem for bounded dynamical systems

Abstract

We show that a continuous map or a continuous flow on ${\mathbb{R}}^n$ with a certain recurrence relation must have a fixed point. Specifically, if there is a compact set W with the property that the forward orbit of every point in ${\mathbb{R}}^n$ intersects W then there is a fixed point in W. Consequently, if the omega limit set of every point is nonempty and uniformly bounded then there is a fixed point.

4 pages, minor clarifications