Connectedness of a class of planar self-affine tiles

Abstract

We consider a class of planar self-affine tiles T that are generated by the lower triangular expanding matrices and the product-form digit sets. We give necessary and sufficient conditions for T to be connected and disk-like. Also for the disconnect case, we give a condition that enumerates the number of connected components of T .

Topological properties of self-similar fractals with one parameter

“…In the first class, we consider a kind of digit sets D ε with a shift ε on the x-coordinates of some digits. We obtain an analogous result to [6]. Theorem 1.1.…”

“…Proof of Theorem 1.1: (i) If |ε| < |p|, by using the same argument as in the proof of Proposition 2.4, we can prove that T ε is disk-like (see also Theorem 3.1 of [6]); If |ε| < |p| n+1 for n ≥ 1, by Proposition 2.4, then G j 1 ···jn are disk-like tiles. Moreover, from Proposition 2.3 and the assumption |ε| ≥ |p| n , it follows that the tiles G j 1 ···jn are either disjoint or meet each other at a single point (see Figure 1).…”

“…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].…”

“…To simplify the analysis on the relations between those two types of "changes", one may fix an expanding matrix A and focus on particular choices of the digit set D. Recently Deng and Lau [6] considered a class of planar selfaffine tiles T that are generated by a lower triangular expanding matrix and productform digit sets. They gave a complete characterization on both connectedness and disk-likeness of T .…”

Height reducing property of polynomials and self-affine tiles

“…The higher dimensional case remains unsolved. In [4], the connectedness of some planar tiles with the other simple arrangement of D has also been considered.…”