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 greater than 1 is a Whitney set; give a geometric sufficient condition for a self-similar arc to be a quasi-arc, and provide an example of a self-similar arc such that any subarc of it fails to be at-quasi-arc for anyt ≥ 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. Supported by Special Funds for Major State Basic Research Projects of China, Morningside Center of Mathematics, NSFC (No. 10241003) and ZJNFS (No. 101026).