Stability of limit expansive systems

Abstract

In this paper, we study the stability of limit expansive systems. More precisely, we prove that if a homeomorphism on a compact metric space is limit expansive and has the shadowing property, then it is topologically Ω-stable. Moreover, a circle homeomorphism is topologically stable if and only if it is limit expansive and has the shadowing property. Furthermore, we show that if a linear operator on a Banach space is limit expansive and has the shadowing property, then it is topologically stable. For a finite dimensional Banach space, the notion of limit expansiveness and topological stability for linear operators are equivalent. Finally, we characterize the notion of Ω-stability for diffeomorphisms on compact smooth manifolds by using the notion of limit expansiveness.