Self-affine tiles in ℝn
Self-affine tiles in ℝn
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DOI：
10.1006/aima.1996.0045
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发表时间：
1996-07
影响因子：
1.7
通讯作者：
J. Lagarias;Yang Wang
中科院分区：
文献类型：
--
作者：
J. Lagarias;Yang Wang
摘要 R n 中的自仿射瓦片是正测度集合 T，其中 A ( T )=∪ d ∈ D ( T + d )，其中 A 是扩展的 n × n 实数矩阵，|det( A )|= m 是整数，D ={ d , d 2 , ..., d m }⊆ R n 是 m 位数字的集合。众所周知，自仿射平铺总是通过平移给出 R n 的平铺。本文扩展了数字集 D 的已知特征，产生自仿射图块。它证明了关于使用这种瓦片可能的 R n 瓦片结构的几个结果，并举例说明了自复制瓦片和一般瓦片之间可能的关系，这澄清了 Kenyon 关于自复制瓦片的结果。
Abstract A self-affine tile in R n is a set T of positive measure with A ( T )=∪ d ∈ D ( T + d ), where A is an expanding n × n real matrix with |det( A )|= m an integer, and D ={ d , d 2 , ..., d m }⊆ R n is a set of m digits. It is known that self-affine tiles always give tilings of R n by translation. This paper extends known characterizations of digit sets D yielding self-affine tiles. It proves several results about the structure of tilings of R n possible using such tiles, and gives examples showing the possible relations between self-replicating tilings and general tilings, which clarify results of Kenyon on self-replicating tilings.