On self-affine tiles that are homeomorphic to a ball  

作　　者：Jorg M.Thuswaldner Shu-Qin Zhang 

机构地区：[1]Mathematics and Statistics,University of Leoben,Leoben 8700,Austria [2]School of Mathematics and Statistics,Zhengzhou University,Zhengzhou 450001,China

出　　处：《Science China Mathematics》2024年第1期45-76,共32页中国科学（数学）（英文版）

基　　金：supported by a grant funded by the Austrian Science Fund and the Russian Science Foundation(Grant No.I 5554);supported by National Natural Science Foundation of China(Grant No.12101566)。

摘　　要：Let M be a 3×3 integer matrix which is expanding in the sense that each of its eigenvalues is greater than 1 in modulus and let D?Z^(3)be a digit set containing|det M|elements.Then the unique nonempty compact set T=T(M,D)defined by the set equation MT=T+D is called an integral self-affine tile if its interior is nonempty.If D is of the form D={0,v,...,(|det M|-1)v},we say that T has a collinear digit set.The present paper is devoted to the topology of integral self-affine tiles with collinear digit sets.In particular,we prove that a large class of these tiles is homeomorphic to a closed 3-dimensional ball.Moreover,we show that in this case,T carries a natural CW complex structure that is defined in terms of the intersections of T with its neighbors in the lattice tiling{T+z:z∈Z^(3)}induced by T.This CW complex structure is isomorphic to the CW complex defined by the truncated octahedron.

关 键 词：self-afine sets tiles and tilings low-dimensional topology truncated octahedron 

分 类 号：O189[理学—数学]