Disk-like tiles and self-affine curves with noncollinear digits

Abstract

Abstract. Let A ∈ M n (Z) be an expanding matrix, D ⊂ Z n a digit set and T = T (A, D) the associated self-affine set. It has been asked by Gröchenig and Haas (1994) that given any expanding matrix A ∈ M 2 (Z), whether there exists a digit set such that T is a connected or disk-like (i.e., homeomorphic to the closed unit disk) tile. With regard to this question, collinear digit sets have been studied in the literature. In this paper, we consider noncollinear digit sets and show the existence of a noncollinear digit set corresponding to each expanding matrix such that T is a connected tile. Moreover, for such digit sets, we give necessary and sufficient conditions for T to be a disk-like tile.

A new class of exceptional self-affine fractals

“…(II)). Then F (T , D) is a class of tiles studied in the proofs of Propositions 3.7, 3.10 in [16]. Note that A, and hence F (T , A), are obtained from C by changing only one digit of C .…”

“…For that, we need to prove thatΓ and F are connected. In [16], it has already been shown that Γ is homeomorphic to the closed unit disk for |p| > 0 and it is a rectangle for p = 0 (see (B) of the proof of Proposition 3.4 in [16]). Therefore, our graph-directed IFS in Section 2 satisfies OSC for the connected open setΓ .…”

“…(a) For the case of reducible c.p., we only sketch the proof, and note that there exists a unimodular matrix P so that PT −1 P −1 is lower triangular [16]. From the considerations in [16], we see that one can take digit sets of the form D = {(i, j) t | 0 ≤ i ≤ r, 0 ≤ j ≤ s} in (II), and in fact, F (T , D) contains a rectangle whose corners can be found explicitly. This enables us to choose C with (I), (II) and eases the tedious computation of ∆ (it is, in fact, this part of the proof that we skip to keep it in a reasonable length).…”

“…This enables us to choose C with (I), (II) and eases the tedious computation of ∆ (it is, in fact, this part of the proof that we skip to keep it in a reasonable length). Also, it was proved in [16] that F (T , D) is homeomorphic to the closed unit disk. Therefore,Γ is connected, i.e.…”

Topological properties of self-similar fractals with one parameter

“…Since the fundamental theory of self-affine tiles was established by Lagarias and Wang ( [13], [14], [15]), there have been considerable interests in the topological structure of self-affine tiles T , including but not limited to the connectedness of T ( [7], [8], [12], [1], [6]), the boundary ∂T ( [2], [19], [22]), or the interior T • of a connected tile T ( [24], [25]). Especially in R 2 , the study on the disk-likeness of T (i.e., the property of being a topological disk) has attracted a lot of attentions ( [5], [16], [23], [11], [6]). For other related works, we refer to [20], [17], [18], [21] and a survey paper [3].…”