-persistently continuum-wise expansive homoclinic classes and recurrent sets

Let p be a hyperbolic periodic point of a diffeomorphism f on a C ∞ closed manifold M. Introducing here the notion of C 1 -persistently continuum-wise expansivity, we show that if the homoclinic class H ( p , f ) of f associated to p is C 1 -persistently continuum-wise expansive then (i) it admits a dominated splitting and (ii) it is hyperbolic provided it satisfies the chain condition. Moreover, we show that the chain recurrent set R ( f ) of f is C 1 -persistently continuum-wise expansive if and only if f satisfies both Axiom A and the no-cycle condition.