Topological stability of anosov flows and their centralizers

Abstract

ANOSOV [l] has shown that an Anosov flow {f’} of a compact manifold !M is structurally stable. Moser [3] states a structural stability for Anosov flows, which is stronger than that of Anosov and gives an outline of the proof (Theorem 3, [3]). The idea of Moser’s proof is very simple (and so interesting) in the sense that he uses only the contraction principle and the estimates of norms of linear operators of Banach spaces. However, his proof seems to have a serious gap (cf. [3], p. 423, F,,‘v,, = v,, and p. 425: first two lines). The authors therefore do not know whether the above Moser’s theorem holds or not. On the other hand, Walters [5] has shown that an Anosov diffeomorphism is topologically stable. In this paperwe show first that an Anosov flow is also topologically stable (Theorem A). The idea of the proof follows that of Moser [3], Walters [5] and Morimoto [2]. Making use of Theorem A we shall next determine the centralizer of an Anosov flow (Theorem B).