In this paper we study structurally stable homoclinic classes. In a natural
way, the structural stability for an individual homoclinic class is defined
through the continuation of periodic points. Since the homoclinic classes is
not innately locally maximal, it is hard to answer whether structurally stable
homoclinic classes are hyperbolic. In this article, we make some progress on
this question. We prove that if a homoclinic class is structurally stable, then
it admits a dominated splitting. Moreover we prove that codimension one
structurally stable classes are hyperbolic. Also, if the diffeomorphism is far
away from homoclinic tangencies, then structurally stable homoclinic classes
are hyperbolic.Comment: arXiv admin note: substantial text overlap with arXiv:1410.430