Chaos for finitely generated semigroup actions

In this paper, we define and study Li-Yorke chaos and distributional chaos along a sequence for finitely generated semigroup actions. Let X be a compact space with metric d and G be a semigroup generated by f1, f2, ... fm which are finitely many continuous mappings from X to itself. Then we show if (X,G) is transitive and there exists a common fixed point for all the above mappings, then (X,G) is chaotic in the sense of Li-Yorke. And we give a sufficient condition for (X,G) to be uniformly distributionally chaotic along a sequence and chaotic in the strong sense of Li-Yorke. At the end of this paper, an example on the one-sided symbolic dynamical system for (X,G) to be chaotic in the strong sense of Li-Yorke and uniformly distributionally chaotic along a sequence is given.