Stably Shadowable Chain Components And Robustly Expansive Homoclinic Classes

Let f be a diffeomorphism on a compact Riemann manifold,and p be a hyperbolicperiodic point of f．Let g be a perturbation of f,denote p_g the continuation of pfor g．Denote C_f(p)the chain component of f that contains p．We say C_f(p)is C~1-stably shadowable if there is a C~1 neighborhood U of f such that for every g∈U,C_g(p_g)has the shadowing property．Denote H_f(P)the homoclinic class of f thatcontains hyperbolic periodic orbit P．We say H_f(P)is C~1 robustly expensive if thereis a C~1 neighborhood U of f such that for every g∈U,H_g(P_g)is expansive．Weprove in this paper that if C_f(p)is C~1-stably shadowable,then C_f(p)is hyperbolic,at the same time,if H_f(P)is robustly expensive,and has shadowing property,thenH_f(P)is hyperbolic．...