Relations among Whitney sets, self-similar arcs and quasi-arcs

We study in this paper some relations among self-similar arcs, Whitney sets and quasi-arcs: we prove that any self-similar arc of dimension than 1 is a Whitney set; give a geometric sufficient condition for a self-similar arc to he a quasi-arc, and provide an example of a self-similar arc such that any subarc of it fails to be a t-quasi-arc for any t greater than or equal to 1, which answers an open question on Whitney sets. We also show that self-similar arcs with the same Hausdorff dimension need not be Lipschitz equivalent.