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2018•Proceedings of the American Mathematical SocietyOpen access

Expansive measures versus Lyapunov exponents

Alma Armijo, Maria José Pacífico

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Abstract

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 C^1 -volume-preserving diffeomorphism that cannot be accumulated by positively measure-expansive diffeomorphisms has a dominated splitting.

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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 C^1 -volume-preserving diffeomorphism that cannot be accumulated by positively measure-expansive diffeomorphisms has a dominated splitting.

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Available abstract

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 C^1 -volume-preserving diffeomorphism that cannot be accumulated by positively measure-expansive diffeomorphisms has a dominated splitting.

Key concepts: Expansive, Ergodic theory, Diffeomorphism, Lyapunov exponent, Measure (data warehouse), Invariant measure, Mathematics, Invariant (physics)

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