Abstract
This paper introduces both the notions of topological transitivity and topological mixing in the general setting of semigroup actions on topological spaces. A discussion on limit behavior of skew-product transformation semigroups is presented. The main purpose is to characterize the lifts and the projections of recurrent points, attractors and Morse decompositions for transformation semigroups associated to skew-product transformation semigroups. The results play a role to the existence of the finest Morse decomposition for control systems and their control flows.
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