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  and Bowen . 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.
"Spectral decomposition and stability of mild expansive systems." Topol. Methods Nonlinear Anal. 56 (1) 63 - 81, 2020. https://doi.org/10.12775/TMNA.2019.119