Chaotic Dynamics of Composition Operators on the Space of Continuous Functions

In this paper, the chaotic dynamics of composition operators on the space of real-valued continuous functions is investigated. It is proved that the hypercyclicity, topologically mixing property, Devaney chaos, frequent hypercyclicity and the specification property of the composition operator are equivalent to each other and are stronger than dense distributional chaos. Moreover, the composition operator C-phi exhibits dense Li-Yorke chaos if and only if it is densely distributionally chaotic, if and only if the symbol phi admits no fixed points. Finally, the long-time behaviors of the composition operator with affine symbol are classified in detail.