Boundary parametrization of planar self-affine tiles with collinear digit set  

作　　者：AKIYAMA Shigeki LORIDANT Benoit 

机构地区：[1]Department of Mathematics,Faculty of Science,Niigata University,Ikarashi 28050 Niigata,9502181,Japan

出　　处：《Science China Mathematics》2010年第9期2173-2194,共22页中国科学：数学（英文版）

基　　金：supported by Japanese Ministry of Education,Culture,Sports,Science and Technology,Grantin Aid for Fundamental Research (Grant No.21540010);the Japanese Society for the Promotion of Science (Grant No.08F08714)

摘　　要：We consider a class of planar self-affine tiles T = M-1 a∈D(T + a) generated by an expanding integral matrix M and a collinear digit set D as follows:M =(0-B 1-A),D = {(00),...,(|B|0-1)}.We give a parametrization S1 →T of the boundary of T with the following standard properties.It is H¨older continuous and associated with a sequence of simple closed polygonal approximations whose vertices lie on T and have algebraic preimages.We derive a new proof that T is homeomorphic to a disk if and only if 2|A| |B + 2|.We consider a class of planar self-affine tiles T = M-1 a∈D(T + a) generated by an expanding integral matrix M and a collinear digit set D as follows:M =(0-B 1-A),D = {(00),...,(|B|0-1)}.We give a parametrization S1 →T of the boundary of T with the following standard properties.It is H¨older continuous and associated with a sequence of simple closed polygonal approximations whose vertices lie on T and have algebraic preimages.We derive a new proof that T is homeomorphic to a disk if and only if 2|A| |B + 2|.

关 键 词：SELF-SIMILAR tile BOUNDARY disk-likeness fractal PARAMETRIZATION 

分 类 号：O186.11[理学—数学]