Topological properties of self-similar fractals with one parameter

In this paper, we study two classes of planar self-similar fractals Tε with a shifting parameter ε. The first one is a class of self-similar tiles by shifting x-coordinates of some digits. We give a detailed discussion on the disk-likeness (i.e., the property of being a topological disk) in terms of ε. We also prove that Tε determines a quasi-periodic tiling if and only if ε is rational. The second one is a class of self-similar sets by shifting diagonal digits. We give a necessary and sufficient condition for Tε to be connected.