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1.
Uniform perfectness of self-affine sets   总被引:2,自引:0,他引:2  
Let be affine maps of Euclidean space with each nonsingular and each contractive. We prove that the self-affine set of is uniformly perfect if it is not a singleton.

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2.
DIMENSIONS OF SELF-AFFINE SETS WITH OVERLAPS   总被引:1,自引:0,他引:1       下载免费PDF全文
The authors develop an algorithm to show that a class of self-affine sets with overlaps canbe viewed as sofic affine-invariant sets without overlaps,thus by using the results of [11]and[10],the Hausdorff and Minkowski dimensions are determined.  相似文献
3.
Self-Affine Sets and Graph-Directed Systems   总被引:1,自引:0,他引:1  
He  Lau  Rao 《Constructive Approximation》2008,19(3):373-397
   Abstract. A self-affine set in R n is a compact set T with A(T)= ∪ d∈ D (T+d) where A is an expanding n× n matrix with integer entries and D ={d 1 , d 2 ,···, d N } ⊂ 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∈ 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 o ≠  ). 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.  相似文献
4.
The authors consider generalized statistically self-affine recursive fractals K with random numbers of subsets on each level. They obtain the Hausdorff dimensions of K without considering whether the subsets on each level are non-overlapping or not. They also give some examples to show that many important sets are the special cases of their models.  相似文献
5.
讨论了三维欧氏空间上的一类自仿射集──广义谢尔宾斯基(Sierpinski)海绵的填充测度(Packing Measure),对φ(t)=tθ,φ(t)=tθ/|logt|及更一般的情况,证明了填充测度Pψ[K(T,D)]为无穷或有限的条件,  相似文献
6.
Using techniques introduced by C. Güntürk, we prove that the attractors of a family of overlapping self-affine iterated function systems contain a neighbourhood of zero for all parameters in a certain range. This corresponds to giving conditions under which a single sequence may serve as a ‘simultaneous ββ-expansion’ of different numbers in different bases.  相似文献
7.
Kaimanovich (2003) [9] introduced the concept of augmented tree on the symbolic space of a self-similar set. It is hyperbolic in the sense of Gromov, and it was shown by Lau and Wang (2009)  [12] that under the open set condition, a self-similar set can be identified with the hyperbolic boundary of the tree. In the paper, we investigate in detail a class of simple augmented trees and the Lipschitz equivalence of such trees. The main purpose is to use this to study the Lipschitz equivalence problem of the totally disconnected self-similar sets which has been undergoing some extensive development recently.  相似文献
8.
The graphs of coordinate functions of space-filling curves such as those described by Peano, Hilbert, Pólya and others, are typical examples of self-affine sets, and their Hausdorff dimensions have been the subject of several articles in the mathematical literature. In the first half of this paper, we describe how the study of dimensions of self-affine sets was motivated, at least in part, by these coordinate functions and their natural generalizations, and review the relevant literature. In the second part, we present new results on the coordinate functions of Pólya's one-parameter family of space-filling curves. We give a lower bound for the Hausdorff dimension of their graphs which is fairly close to the box-counting dimension. Our techniques are largely probabilistic. The fact that the exact dimension remains elusive seems to indicate the need for further work in the area of self-affine sets.  相似文献
9.
Recently Lipschitz equivalence of self‐similar sets on has been studied extensively in the literature. However for self‐affine sets the problem is more awkward and there are very few results. In this paper, we introduce a w‐Lipschitz equivalence by repacing the Euclidean norm with a pseudo‐norm w. Under the open set condition, we prove that any two totally disconnected integral self‐affine sets with a common matrix are w‐Lipschitz equivalent if and only if their digit sets have equal cardinality. The main methods used are the technique of pseudo‐norm and Gromov hyperbolic graph theory on iterated function systems.  相似文献
10.
For a d ×d expanding matrix A, we de.ne a pseudo‐norm w (x) in terms of A and use this pseudo‐norm (instead of the Euclidean norm) to define the Hausdorff measure and the Hausdorff dimension dimw H E for subsets E in R d . We show that this new approach gives convenient estimations to the classical Hausdorff dimension dimw H E, and in the case that the eigenvalues of A have the same modulus, then dimw H E and dimH E coincide. This setup is particularly useful to study self‐affine sets T generated by ϕj (x) = A–1(x +dj), dj ∈ R d , j = 1, …, N. We use it to investigate the fractality of T for the case that {ϕj }N j =1 satisfying the open set condition as well as the cases without the open set condition. We extend some well‐known results in the self‐similar sets to the self‐af.ne sets. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献
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