弱双曲轨道的稳定流形刻画及其应用

The stable manifold plays an important role in the study of differentiable dynamics. For a diffeomorphism f, the stable manifolds of a hyperbolic set have the following property: there exists a positive constant ε > 0 such that for any x in the hyperbolic set, the set {y : d(fⁿx, fⁿy) 6 ε, ∀n > 0} is contained in the stable manifold of x. This paper extends this result to the following conclusion: if an invariant set Λ admits a dominated splitting TΛM = E ⊕ F, then for any positive integer m and constant η > 0, there exists ε > 0 such that when a point x ∈ Λ satisfies (Formular Presented) then the set {y : d(fⁿx, fⁿy) 6 ε, ∀n > 0} is contained in the stable manifold of x. In addition to proving this theorem, this paper also presents its application on invariant sets with local star properties.