Self-affine sets and graph-directed systems
Publication in refereed journal

香港中文大學研究人員

引用次數
替代計量分析
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其它資訊
摘要A self-affine set in R-n is a compact set T with A(T) = U-dis an element ofD(T+d) where A is an expanding n x n matrix with integer entries and D = {d(1), d(2), . . ., d(N)} subset of Z(n) is an N-digit set. For the case N = \det(A)\ the set T has been studied in great detail in the context of self-affine tiles. Our main interest in this paper is to consider the case N > \det(A)\, but the theorems and proofs apply to all the N. The self-affine sets arise naturally in fractal geometry and, moreover, they are the support of the scaling functions in wavelet theory. The main difficulty in studying such sets is that the pieces T + d, d is an element of D, overlap and it is harder to trace the iteration. For this we construct a new graph-directed system to determine whether such a set T will have a nonvoid interior, and to use the system to calculate the dimension of T or its boundary (if T-0 not equal circle divide). By using this setup we also show that the Lebesgue measure of such T is a rational number, in contrast to the case where, for a self-affine tile, it is an integer.
著者He XG, Lau KS, Rao H
期刊名稱Constructive Approximation
出版年份2003
月份1
日期1
卷號19
期次3
出版社SPRINGER-VERLAG
頁次373 - 397
國際標準期刊號0176-4276
電子國際標準期刊號1432-0940
語言英式英語
關鍵詞attractor; boundary; contraction; graph-directed construction; Hausdorff dimension; iterated function system; overlapping; self-affine sets; self-similar sets; tiles
Web of Science 學科類別Mathematics; MATHEMATICS

上次更新時間 2020-22-11 於 01:23