POSITIVELY MEASURE EXPANSIVE AND EXPANDING

Abstract

Abstract. We show that C 1 -generically, a differentiable map is posi-tively measure expansive if and only if it is expanding. 1. IntroductionLet M be a compact connected C ∞ Riemannian manifold without bound-ary and C 1 (M) the space of differentiable maps of M endowed with the C 1 -topology. Denote by d the distance on M induced from the Riemannian metrick·k on the tangent bundle TM. Given x ∈ M and δ > 0, define the dynamicalδ-ball, Γ δ (x) = {y ∈ M : d(f i (x),f i (y)) ≤ δ for all i ≥ 0}. Let µ be a Borelprobability measure which is not necessary f-invariant. Let f ∈ C 1 (M). Wesay that f is positively measure expansive (or, positively µ-expansive) if there isδ > 0 (called expansive constant) such that for all x ∈ M, µ(Γ δ (x)) = 0. It isknown that if f is positively expansive, then f is open and locally one-to-one,that is, f is a local homeomorphism since M is a manifold without boundary.Since M is connected, it can be checked that the set of periodic points, P(f),of f is dense (see [4]).We say that f is expanding if there are constants C > 0 and λ > 1 suchthat for any v ∈ T