首页|On self-affine tiles that are homeomorphic to a ball
On self-affine tiles that are homeomorphic to a ball
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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 C Z3 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 ∈ Z3} induced by T.This CW complex structure is isomorphic to the CW complex defined by the truncated octahedron.
self-affine setstiles and tilingslow-dimensional topologytruncated octahedron
J?rg M.Thuswaldner、Shu-Qin Zhang
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Chair of Mathematics and Statistics,University of Leoben,Leoben 8700,Austria
School of Mathematics and Statistics,Zhengzhou University,Zhengzhou 450001,China
Austrian Science Fund and the Russian Science FoundationNational Natural Science Foundation of China
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