Relations among Whitney sets, self-similar arcs and quasi-arcs (Q1424111)

The authors study some relations among self-similar arcs, Whitney sets and quasi-arcs and give an affirmative answer to the open problem posed by Norton in 1989. They prove that any self-similar arc of Hausdorff dimension greater than 1 is a Whitney set and there is a self-similar arc \(\gamma\) with \(\dim_H\gamma > 1\) such that every subarc \(\eta\) of \(\gamma\) fails to be a \(t\)-quasi-arc for any \(t\geq 1.\) They also give a sufficient condition for a self-similar arc to be a quasi-arc and show that self-similar arcs with the same Hausdorff dimension need not be Lipschitz equivalent.