C1-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 C1-persistently continuum-wise expansivity, we show that if the homoclinic class H(p,f) of f associated to p is C1-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 C1-persistently continuum-wise expansive if and only if f satisfies both Axiom A and the no-cycle condition.