Abstract
In the study, we consider continuum-wise expansiveness for the homoclinic class of a kind of C^{1}-robustly expansive dynamical system. First, we show that if the homoclinic class H(p, f), which contains a hyperbolic periodic point p, is R-robustly continuum-wise expansive, then it is hyperbolic. For a vector field, if the homoclinic class H(gamma , X) does not include singularities and is R-robustly continuum-wise expansive, then it is hyperbolic.
Highlights
1.1 Continuum-wise expansiveness for diffeomorphisms Let M be a closed connected smooth Riemannian manifold
⎨expp ⎩f (x) exp–p1 (x) if x ∈ B 0/4(p) ∩ Np,r, if x ∈/ B 0 (p) ∩ Np,r, where B (x) is a closed ball in M center at x ∈ M with radius > 0, Fp(Xt, r, T/2) = {Xt(y) : y ∈ Nx,r and 0 ≤ t ≤ T}, and g : Np,r → Np is the Poincaré map defined by Yt
Proof Since HX(γ ) ∩ Sing(X), we prove that if H(γ, X) is R-robustly continuum-wise expansive, every η ∈ HX(γ ) ∩ Per(X) is hyperbolic
Summary
1.1 Continuum-wise expansiveness for diffeomorphisms Let M be a closed connected smooth Riemannian manifold. We consider the hyperbolicity of the homoclinic class H(γ , X) under a type of C1-robustly continuum-wise expansiveness.
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