Entropy-Expansiveness Of C~1 Generic Diffeomorphisms

Let M be a d dimensional closed smooth Riemannian manifold,d≥2,f:M→M be a C¹diffeomorphism,and H（p）be a homoclinic class associated with hyperbolic periodic point p.In this paper,we prove that if f|_H（p）is entropy-expansive,then H（p）has a Df-invariant dominated splitting of the form T_H（p）M=E⊕F₁⊕F₂⊕···⊕F_k⊕G（k∈N）,where all F_l（l=1,···,k）are one-dimensional and not hyperbolic and dim E=i（p）,dim G=dim M-j（p）,k=j（p）-i（p）,where i（p）and j（p）are the minimum and maximum indexes of hyperbolic periodic points in H（p）respectively.Furthermore,if H（p）is isolated,then the above dominated splitting has the partial hyperbolicity,that is,E is uniformly contracting and G is uniformly expanding.