C1-STABLY EXPANSIVE SETS FOR FLOWS

Abstract. Let X be a C1 vector field on a closed C ∞ manifold M. We introduce the concept of C1 stable expansivity for compact Xt-invariant set, and use a flow-version of Mane’s results (Lemma II.3 in ”Mane, R. An ergodic closing lemma. Ann. of Math. (2) 116 (1982), no. 3, 503–540”) on uni-formly hyperbolic families of periodic linear differential equations in order to get the hyperbolicity for the C1 stably expansive homoclinic class HX(γ) of a hyperbolic periodic orbit γ. 1. Main results Let X be a C1 vector field on a closed C ∞ manifold M, and X 1(M) be the set of C1 vector field on M endowed with C1 topology. Denoted E(M) by the set of expansive vector fields on M. Given a vector field X ∈ X 1(M), we denote Sing(X) by the set of singularities of X; PO(Xt) by the set of periodic orbits of Xt; R(X) by the chain recurrent set of X. Let γ be a hyperbolic periodic orbit of Xt. W s(γ) and Wu(γ) denote the stable and unstable manifolds of γ; HX(γ) denotes the transversal homoclinic class of X associated with γ, i.e., HX(γ) = W s(γ)tWu(γ); CX(γ) denotes the chain component of X containing γ; Pt denotes the linear Poincaré flow defined on the normal bundle to X over M −Sing(X). For these definitions as well as hyperbolicity and dominated splitting of the linear Poincare ́ flow, see [2, 19]. The following result is from [12]