Periodically Expansive Homoclinic Classes

In this paper we introduce the notion of periodically expansiveness and discuss the homoclinic classes exhibit the property in a persistent way. More precisely, we prove that if a homoclinic class H(p, f) of a diffeomorphisms f is C ¹-persistently periodically expansive then it admits a dominated splitting ${E \bigoplus F}$ with dim?(E)?=?index?(p). We also prove that C ¹-generically any locally maximal periodically expansive homoclinic class is hyperbolic.