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- STABLY EXPANSIVE SETS FOR FLOWS

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 uniformly 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 C vector field on a closed C∞ manifold M , and X (M) be the set of C vector field on M endowed with C topology. Denoted E(M) by the set of expansive vector fields on M . Given a vector field X ∈ X (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 (γ) and W(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 Poincaré flow, see [2, 19]. The following result is from [12].