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SHADES OF HYPERBOLICITY FOR HAMILTONIANS
, 2012
"... We prove that a Hamiltonian system H ∈ C2(M,R) is globally hyperbolic if any of the following statements holds: H is robustly topologically stable; H is stably shadowable; H is stably expansive; and H has the stable weak specification property. Moreover, we prove that, for a C2-generic Hamiltonian ..."
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We prove that a Hamiltonian system H ∈ C2(M,R) is globally hyperbolic if any of the following statements holds: H is robustly topologically stable; H is stably shadowable; H is stably expansive; and H has the stable weak specification property. Moreover, we prove that, for a C2-generic Hamiltonian H, the union of the partially hyperbolic regular energy hypersurfaces and the closed elliptic orbits, forms a dense subset of M. As a consequence, any robustly transitive regular energy hypersurface of a C2-Hamiltonian is partially hyperbolic. Finally, we prove that stably weakly shadowable regular energy hypersurfaces are partially hyperbolic.
STRUCTURAL STABILITY OF ORBITAL INVERSE SHADOWING VECTOR
"... Abstract. In this paper we give a characterization of the structurally stable vector fields via the notion of orbital inverse shadowing. More precisely it is proved that the C1 interior of the set of C1 vector fields with the orbital inverse shadowing property coincides with the set of structurally ..."
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Abstract. In this paper we give a characterization of the structurally stable vector fields via the notion of orbital inverse shadowing. More precisely it is proved that the C1 interior of the set of C1 vector fields with the orbital inverse shadowing property coincides with the set of structurally stable vector fields on a compact smooth manifold. This fact improves the result obtained by K. Moriyasu, K. Sakai and N. Sumi in [13]. 1.
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 hy ..."
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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].
STRUCTURAL STABILITY OF VECTOR FIELDS WITH ORBITAL INVERSE SHADOWING
"... Abstract. In this paper, we give a characterization of the structurally stable vector fields via the notion of orbital inverse shadowing. More pre-cisely, it is proved that the C1 interior of the set of C1 vector fields with the orbital inverse shadowing property coincides with the set of struc-tura ..."
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Abstract. In this paper, we give a characterization of the structurally stable vector fields via the notion of orbital inverse shadowing. More pre-cisely, it is proved that the C1 interior of the set of C1 vector fields with the orbital inverse shadowing property coincides with the set of struc-turally stable vector fields. This fact improves the main result obtained by K. Moriyasu et al. in [15]. 1.
