Spectral decomposition and stability of mild expansive systems

In this paper we introduce another type of expansiveness, called mild expansiveness, for homeomorphisms on compact metric spaces, and present a topological version of the spectral decomposition theorem by Smale \cite{Sm} and Bowen \cite{Bo}. Moreover we study the topological stability of mild expansive homeomorphisms, and characterize the mild expansive diffeomorphisms on compact smooth manifolds using the notion of $\Omega$-stability.