Compact Orbits and Topological Sensitivity on Locally Compact Spaces

We introduce a cover-based formulation of topological sensitivity and sensitivity at a point for continuous self-maps on locally compact spaces, extending the classical metric framework. Using the one-point compactification, we analyze the basin of attraction of infinity and relate it to escaping dynamics. We study the set K(f) consisting of points whose forward orbits are contained in a compact subset of the phase space, establishing its fundamental topological properties under suitable assumptions on the map f. In particular, we show that for proper maps, K(f) coincides with the complement of the escaping set. Under additional hypotheses on f, we prove that the boundary of the set of points with compact orbits, the boundary of the basin of attraction of infinity, and the set of sensitive points coincide. This provides a topological generalization of the classical dichotomy between Fatou and Julia sets in complex dynamics.