Characterization of stable manifolds of orbits with weak hyperbolicity and its application

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 断言: there exists a positive constant varepsilon>0 such that for any x in the hyperbolic set, the set y: d(f^nx, f^n y)≤ varepsilon, ∀ ngeq 0is 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 T_Λ M=E⊕ F, then for any positive integer m and constant η>0, there exists varepsilon>0 such that when a point x∈Λ satisfiesłimsup_nto+∞frac1n