  We discuss different regularities on stable/unstable holonomies of
cw-hyperbolic homeomorphisms and prove that if a cw-hyperbolic homeomorphism
has continuous joint stable/unstable holonomies, then it has a dense set of
periodic points in its non-wandering set. For that, we prove that the
hyperbolic cw-metric (introduced in [9]) can be adapted to be self-similar (as
in [6]) and, in this case, continuous joint stable/unstable holonomies are
pseudo-isometric. We also prove transitivity of cw-hyperbolic homeomorphisms
assuming that the stable/unstable holonomies are isometric. In the case the
ambient space is a surface, we prove that a cw$_F$-hyperbolic homeomorphism has
continuous joint stable/unstable holonomies when every bi-asymptotic sector is
regular.
