Self-Affine Sets and Graph-Directed Systems

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.     

自仿测度的非谱准则

设μ_(M,D)是由仿射迭代函数系{φ_d(x)=M~(-1)(x+d)}_(d∈D)唯一确定的自仿测度,它的谱性或非谱性与Hilbert空间L~2(μ_(M,D))中正交指数基(也称为Fourier基)的存在性有着直接的关系.近年来自仿测度μ_(M,D)的谱性或非谱性问题的研究受到人们普遍的关注.本文给出了判定自仿测度μ_(M,D)非谱性的几个充分条件,所得结果改进推广Dutkay,Jorgensen等人的非谱准则.     

Rigidity Theorem for Self-Affine Arcs

Doklady Mathematics - It has been known for more than a decade that, if a self-similar arc $$\gamma $$ can be shifted along itself by similarity maps that are arbitrarily close to identity, then...     

Self-Affine Sets and Graph-Directed Systems

   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.     

DIMENSIONS OF SELF-AFFINE SETS WITH OVERLAPS

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.     

On the Connectedness of Self-Affine Tiles

Let T be a self-affine tile in Rⁿ defined by an integral expandingmatrix A and a digit set D. The paper gives a necessary andsufficient condition for the connectedness of T. The conditioncan be checked algebraically via the characteristic polynomialof A. Through the use of this, it is shown that in R², for anyintegral expanding matrix A, there exists a digit set D suchthat the corresponding tile T is connected. This answers a questionof Bandt and Gelbrich. Some partial results for the higher-dimensionalcases are also given.     

Self-Affine Fractal Functions and Wavelet Series

We consider functions represented by series ∑_g  G c_gψ(g ^− 1(x)) of wavelet-type, where G is a group generated by affine functions L₁,…,L_n and ψ is piecewise affine. By means of those functions we characterize the class of self-affine fractal functions, previously studied by Barnsley et al. We compute their global and local Hölder exponents and investigate points of non-differentiability. Wavelet-representations for various continuous nowhere differentiable and singular functions are presented. Another application is the construction of functions with prescribed local Hölder exponents at each point.     

Disk-Like Self-Affine Tiles in R 2

We give simple necessary and sufficient conditions for self-affine tiles in R ² to be homeomorphic to a disk. Received October 10, 2000, and in revised form February 16, 2001, and April 25, 2001. Online publication July 25, 2001.     

On Lebesgue Measure of Integral Self-Affine Sets

Let A be an expanding integer n×n matrix and D be a finite subset of ? ⁿ . The self-affine set T=T(A,D) is the unique compact set satisfying the equality \(A(T)=\bigcup_{d\in D}(T+d)\). We present an effective algorithm to compute the Lebesgue measure of the self-affine set T, the measure of the intersection T∩(T+u) for u∈? ⁿ , and the measure of the intersection of self-affine sets T(A,D ₁)∩T(A,D ₂) for different sets D ₁, D ₂?? ⁿ .     

Expanding Polynomials and Connectedness of Self-Affine Tiles

Little is known about the connectedness of self-affine tiles in ${\Bbb R}^n$. In this note we consider this property on the self-affine tiles that are generated by consecutive collinear digit sets. By using an algebraic criterion, we call it the {\it height reducing property}, on expanding polynomials (i.e., all the roots have moduli $ > 1$), we show that all such tiles in ${\Bbb R}^n, n \leq 3$, are connected. The problem is still unsolved for higher dimensions. For this we make another investigation on this algebraic criterion. We improve a result of Garsia concerning the heights of expanding polynomials. The new result has its own interest from an algebraic point of view and also gives further insight to the connectedness problem.     

Measure of Self-Affine Sets and Associated Densities

Let \(B\) be an \(n\times n\) real expanding matrix and \(\mathcal {D}\) be a finite subset of \(\mathbb {R}^n\) with \(0\in \mathcal {D}\) . The self-affine set \(K=K(B,\mathcal {D})\) is the unique compact set satisfying the set-valued equation \(BK=\bigcup _{d\in \mathcal {D}}(K+d)\) . In the case where \(\#\mathcal D=|\det B|,\) we relate the Lebesgue measure of \(K(B,\mathcal {D})\) to the upper Beurling density of the associated measure \(\mu =\lim _{s\rightarrow \infty }\sum _{\ell _0, \ldots ,\ell _{s-1}\in \mathcal {D}}\delta _{\ell _0+B\ell _1+\cdots +B^{s-1}\ell _{s-1}}.\) If, on the other hand, \(\#\mathcal D<|\det B|\) and \(B\) is a similarity matrix, we relate the Hausdorff measure \(\mathcal {H}^s(K)\) , where \(s\) is the similarity dimension of \(K\) , to a corresponding notion of upper density for the measure \(\mu \) .     

Classification of Integral Expanding Matrices and Self-Affine Tiles

   Abstract. Let T be a self-affine tile that is generated by an expanding integral matrix A and a digit set D . It is known that many properties of T are invariant under the Z -similarity of the matrix A . In [LW1] Lagarias and Wang showed that if A is a 2 × 2 expanding matrix with |det(A)| = 2 , then the Z -similar class is uniquely determined by the characteristic polynomial of A . This is not true if |det(A)| > 2. In this paper we give complete classifications of the Z -similar classes for the cases |det(A)| =3, 4, 5 . We then make use of the classification for |det(A)| =3 to consider the digit set D of the tile and show that μ(T) >0 if and only if D is a standard digit set. This reinforces the conjecture in [LW3] on this.     

Construction of Wavelet Sets Using Integral Self-Affine Multi-tiles

Gabardo and Yu first considered using integral self-affine tiles in the Fourier domain to construct wavelet sets and they produced a class of compact wavelet sets with certain self-similarity properties. In this paper, we generalize their results to the integral self-affine multi-tiles setting. We characterize some analytic properties of integral self-affine multi-tiles under certain conditions. We also consider the problem of constructing (multi)wavelet sets using integral self-affine multi-tiles.     

A Characterization on the Spectra of Self-Affine Measures

Journal of Fourier Analysis and Applications - A discrete set $$\Lambda \subseteq {\mathbb {R}}^d$$ is called a spectrum for the probability measure $$\mu $$ if the family of functions $$\{e^{2 \pi...     

Copulae,Self-Affine Functions,and Fractal Dimensions

We use the fact that the functions defined on the unit interval whose graphs support a copula are those that are Lebesgue-measure-preserving in order to characterize self-affine functions whose graphs are the support of a copula. This result allows computation of the Hausdorff, packing, and box-counting dimensions. The discussion is applied to classic examples such as the Peano and Hilbert curves, and the results are extended to discontinuous self-affine functions.     

Properties of Some Overlapping Self-Similar and Some Self-Affine Measures

We generalize theorems of Peres and Solomyak about the abso- lute continuity resp. singularity of Bernoulli convolutions ([19], [16], [17]) to a broader class of self-similar measures on the real line. Using the dimension the- ory of ergodic measures (see [11] and [2]) we find a formula for the dimension of certain self-affine measures in terms of the dimension of the above mentioned self- similar measures. Combining these results we show the identity of Hausdorff and box-counting dimension of a special class of self-affine sets.     

一类平面自仿射tiles的边界

考虑由扩张矩阵A=(?)及数字集D=(?):0≤i≤|p|-1,O≤j≤|q|一1(?)生成的自仿射tiles集T=T(A,D),其中p,q∈Z,|p|≥2,|q|≥2,通过对T中的元素进行分析,得到了计算T的边界的方法.     

自仿函数分数阶导数的分形维数

研究一类自仿函数的分数阶导数,获得了自仿函数的Weyl-Marchaud分数阶导数的图像盒维数,证明了分数阶导数的阶与分形维数之间的线性关系.     

Spectral Self-Affine Measures on the Generalized Three Sierpinski Gasket

The self-affine measure μM,Dassociated with an iterated function system{φd(x)=M~(-1)(x + d)}_(d∈D) is uniquely determined. It only depends upon an expanding matrix M and a finite digit set D. In the present paper we give some sufficient conditions for finite and infinite families of orthogonal exponentials. Such research is necessary to further understanding the non-spectral and spectral of μM,D. As an application,we show that the L~2(μM,D) space has infinite families of orthogonal exponentials on the generalized three Sierpinski gasket. We then consider the spectra of a class of self-affine measures which extends several known conclusions in a simple manner.     

随机自仿射集的Hausdorff维数估计

定义了一类广泛的随机自仿射集,得到了此类集合的Ｈａｕｓｄｏｒｆｆ维数估计．此前的随机自相似（包括Ｇｒａｆ,Ｍａｕｌｄｉｎ与Ｆａｌｃｏｎｅｒ等定义的随机自相似情形）和Ｆａｌｃｏｎｅｒ定义的（严格）自仿射以及作者定义的μ 统计自仿射情形均成为该文结果的特例．