Radix expansions and connectedness of planar self-affine fractals
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Radix expansions and connectedness of planar self-affine fractals Lian Wang1 · King-Shun Leung2 Received: 10 October 2019 / Accepted: 17 August 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract Let A be an expanding matrix with characteristic polynomial f (x) = x 2 + px + 3 and D = {0, v, v + k Av} be a digit set where , k ∈ Z, v ∈ R2 so that {v, Av} is linearly independent. It is well known that there exists a unique self-affine fractal T satisfying AT = T + D. In this paper, we give a complete characterization for the connectedness of T by using radix expansion. Keywords Self-affine set · Connectedness · Radix expansion · Neighbor-generating scheme Mathematics Subject Classification Primary 28A80; Secondary 52C20 · 54D05
1 Introduction Denote by Mn (Z) the set of n × n matrices with integer entries. Let A ∈ Mn (Z) be an expanding matrix (i.e., all eigenvalues of A have moduli strictly larger than 1). Let D = {d1 , d2 , . . . , dm } ⊂ Rn be an m-digit set where m = | det(A)|. It is well known that there exists a unique nonempty compact set T := T (A, D) [7] satisfying T = A−1 (T + D). We call T a self-affine set, and a self-affine tile, if moreover, it has nonvoid interior. Usually, we also use radix expansion to express the elements of T as follows: Communicated by H. Bruin.
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King-Shun Leung [email protected] Lian Wang [email protected]
1
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, People’s Republic of China
2
Department of Mathematics and Information Technology, The Education University of Hong Kong, Tai Po, Hong Kong
123
L. Wang, K.-S. Leung
T =
∞
−i
A d ji : d ji ∈ D .
i=1
The study on the topological properties of self-affine sets has been an interesting topic in the literature (please see [1,5,6,11–18]). Gröchenig and Haas [8] raised the question that given an expanding integer matrix A, whether there exists a digit set D such that T (A, D) is a connected tile. Hacon et al. [10] proved that any self-affine tile T (A, D) with 2-digit set is always pathwise connected. Until 2000, Kirat and Lau [11] provided a criterion for self-affine tiles to be connected. On R2 , Bandt and Gelbrich [2] first considered the disk-like property of planar self-affine tile (the property of being homeomorphic to a closed disk) for | det(A)| = 2 or 3. Bandt and Wang [3] gave a topological criterion for self-affine tiles to be disk-like. After that, Leung and Lau [12] obtained a simple algebraic condition on the disk-likeness of self-affine tiles generated by a consecutive collinear digit set D = {0, 1, . . . , m − 1}v where v ∈ Zn . More recently, Conner and Thuswaldner [4] and Deng et al. [6] studied the ball-like property of self-affine tiles in R3 . However, there are limited results about the connectedness of self-affine set T (A, D) that are generated by non-consecutive or non-collinear digit sets [5,15,17,18]. On R2 , when | det(A)| = 3, Leung and Luo [13,14] solved the non-consecutive collinear case D = {0, v, v} where > 2 and non-collinear
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