In this paper we investigate the relation between measure-expansiveness and hyperbolicity. We prove that non-atomic invariant ergodic measures with all of their Lyapunov exponents positive are positively measure-expansive. We also prove that local diffeomorphisms robustly positively measure-expansive are expanding. Finally, we prove that a C 1-volume-preserving diffeomorphism that cannot be accumulated by positively measure-expansive diffeomorphisms has a dominated splitting.
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Expansive Measures Versus Lyapunov Exponents
