C~1-stably (L)_p shadowable homoclinic classes

摘要

Let p be a hyperbolic saddle of diffeomorphism f on closed manifold M and H(p,f) be the homoclinic class associated with it. In this article, we introduce the notion of C~1-stabLy (L)_p shadowing and prove that if/is C~1-stabLy C(L)_p shadowable on a homoclinic class H{p,f) then, H(p,f) has a dominated splitting. Moreover, we prove that if/is C'-stably Lipschitz (L)_p, shadowable on H(p,f) and H(p,f)-germ of f is expansive then the homoclinic class is hyperbolic.