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

Stable/unstable holonomies, density of periodic points, and transitivity for continuum-wise hyperbolic homeomorphisms