On the limit set maps in semidynamical systems

In this paper, we discuss the semicontinuities of the orbital and limit set maps, and investigate their relationships with the stabilities of orbits. First we prove that if a point is positively Zhukovskij stable, then the limit set maps are upper semicontinuous at that point. Then we show that if the limit sets of a point are stable (or eventually stable or uniformly eventually stable), then the corresponding limit set maps are upper semicontinuous at that point. Furthermore, we give several sufficient conditions to guarantee that the limit set maps are lower semicontinuous.