Abstract. In this paper, we show that if a nontrivial transitiveset is C 1 -stably measure expansive, then it admits a dominatedsplitting. 1. IntroductionLet Ma compact connected C 1 Riemannian manifold without bound-ary, and Di 1 (M) be the space of di eomorphisms of Mendowed withthe C 1 -topology. Denote by dthe distance on M induced from theRiemannian metric kkon the tangent bundle TM:Let f2Di 1 (M):The notion of expansiveness was introduced by Utz [8] in the middleof the twentieth century. Roughly speaking, a system is expansive if twoorbits cannot remain close to each other under the action of the system.This notion is very important in the context of the theory of dynamicalsystems, for example, the proof of the existence of Markov partitions.As pointed out by Morales [6], in light of the rich consequences of expan-siveness in the dynamics of a system, it is natural to consider anothernotions of expansiveness.Let us start with the di erent de nitions of expansiveness we shalldeal with. Given x2Mand >0, de ne the dynamical -ball, the set