四分Cantor测度的谱性质

自仿测度μM,D谱性质的研究始于四分Cantor测度μ4(即M=4,D={0,2}的情形).在长期从事谱集研究的基础上,Jorgensen和Pedersen在1998年首次发现μ4是一个具有谱性质的分形测度,其谱Λ(M,S)与和谐对(M~(-1)D,S)密切相关,其中S={0,1}.近年来的研究表明,对于某些奇数l,数乘集合lΛ(M,S)也是测度μ4的谱.这使得测度μ4的一些谱具有较强的稀疏性.本文重点对具有上述性质的奇数l进行讨论.利用数论中同余关系和有限群中元素的阶的性质,得到当l分别为素数、素数幂和素数乘积时,lΛ(M,S)为谱的判别依据,改进推广Dutkay等人的工作.     

Duality relation on spectra of self‐affine measures

The present paper establishes a duality relation for the spectra of self‐affine measures. This is done under the condition of compatible pair and is motivated by a duality conjecture of Dutkay and Jorgensen on the spectrality of self‐affine measures. For the spectral self‐affine measure, we first obtain a structural property of spectra which indicates that one can get new spectra from old ones. We then establish a duality property for the spectra which confirms the conjecture in a certain case.     

关于Strichartz的一个谱对准则

研究了与压缩迭代函数系和扩张迭代函数系相关的自仿测度的谱性质.在和谐对的条件下,分别确定了谱对形成的一些充分条件和必要条件.首先,给出了Strichartz谱对准则的几个等价形式.其次,得到了这个谱对成立的两个必要条件.最后,提供了Strichartz谱对准则的一个严格而详细的证明.     

Spectrality of planar self-affine measures with two-element digit set

The iterated function system with two element digit set is the simplest case and the most important case in the study of self affine measures.The one dimensional case corresponds to the Bernoulli convolution whose spectral property is understandable.The higher dimensional analogue is not known,for which two conjectures about the spectrality and the non spectrality remain open.In the present paper,we consider the spectrality and non spectrality of planar self affine measures with two element digit set.We give a method to deal with the two dimensional case,and clarify the spectrality and non spectrality of a class of planar self affine measures.The result here provides some supportive evidence to the two related conjectures.     

Spectrality of a class of self-affine measures with decomposable digit sets

The self-affine measure associated with an expanding matrix and a finite digit set is uniquely determined by the self-affine identity with equal weight.The spectral and non-spectral problems on the selfaffine measures have some surprising connections with a number of areas in mathematics,and have been received much attention in recent years.In the present paper,we shall determine the spectrality and non-spectrality of a class of self-affine measures with decomposable digit sets.We present a method to deal with such case,and clarify the spectrality and non-spectrality of a class of self-affine measures by applying this method.     

Singularity of certain self-affine measures

The self-affine measure associated with an iterated function system and a weight is uniquely determined. The problem of determining whether a self-affine measure is absolutely continuous or singular has been studied extensively in recent years. In the present paper we consider the singularity of certain self-affine measures. We obtain a sufficient condition for such measures being singular. Two applications of this result are given, which extend several known results in a simple manner.     

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.     

Integral Self-affiine Tiles of Bandt＇s Model

Integral self-affine tiling of Bandt＇s model is a generalization of the integral self-affine tiling. Using ergodic theory, we show that the Lebesgue measure of the tile is a rational number where the denominator equals to the order of the associate symmetry group. We apply the result to the study of the Levy Dragon.     

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.     

On the spectra of a Cantor measure

We analyze all orthonormal bases of exponentials on the Cantor set defined by Jorgensen and Pedersen in J. Anal. Math. 75 (1998) 185-228. A complete characterization for all maximal sets of orthogonal exponentials is obtained by establishing a one-to-one correspondence with the spectral labelings of the infinite binary tree. With the help of this characterization we obtain a sufficient condition for a spectral labeling to generate a spectrum (an orthonormal basis). This result not only provides us an easy and efficient way to construct various of new spectra for the Cantor measure but also extends many previous results in the literature. In fact, most known examples of orthonormal bases of exponentials correspond to spectral labelings satisfying this sufficient condition. We also obtain two new conditions for a labeling tree to generate a spectrum when other digits (digits not necessarily in {0,1,2,3}) are used in the base 4 expansion of integers and when bad branches are allowed in the spectral labeling. These new conditions yield new examples of spectra and in particular lead to a surprizing example which shows that a maximal set of orthogonal exponentials is not necessarily an orthonormal basis.     

非谱自仿测度下正交指数函数系的基数

设μ_(M,D)是由扩张矩阵M∈M_n(Z)和有限数字集D?Z~n通过仿射迭代函数系统{φ_d(x)=M~(-1)(x+d)}_(d∈D)唯一确定的自仿测度,它的非谱性与相应的平方可积函数构成的Hilbert空间L~2(μ_(M,D))中正交指数函数系的有限性或无限性密切相关.通过对数字集D的符号函数m_D(x)的零点集合Z(m_D)的特征分析以及其中非零中间点(即坐标为0或1/2的点)和非中间点的性质应用,得到了非谱自仿测度下正交指数函数系基数的一个更为精确的估计,改进推广了Dutkay,Jorgensen等人的相关结果.     

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.     

Absolute continuity of vector-valued self-affine measures

This paper is devoted to the absolute continuity of (scalar-valued or vector-valued) self-affine measures and their properties on the boundary of an invariant set. We first extend the definition of WSC to self-affine IFS, and then obtain a necessary and sufficient condition for the vector-valued self-affine measures to be absolutely continuous with respect to the Lebesgue measure. In addition, we prove that, for any IFS and any invariant open set V, the corresponding (scalar-valued or vector-valued) invariant measure is supported either in V or in ∂V.     

Spectral self-affine measures on the planar Sierpinski family

The present research will concentrate on the topic of Fourier analysis on fractals. It mainly deals with the problem of determining spectral self-affine measures on the typical fractals: the planar Sierpinski family. The previous researches on this subject have led to the problem within the possible fifteen cases. We shall show that among the fifteen cases, the nine cases correspond to the spectral measures, and reduce the remnant six cases to the three cases. Thus, for a large class of such measures, their spectrality and non-spectrality are clear. Moreover, an explicit formula for the existent spectrum of a spectral measure is obtained. We also give a concluding remark on the remnant three cases.     

Spectral self-affine measures with prime determinant

The self-affine measure $\mu _{M,D}$ relating to an expanding matrix $M\in M_{n}(\mathbb Z )$ and a finite digit set $D\subset \mathbb Z ^n$ is a unique probability measure satisfying the self-affine identity with equal weight. In the present paper, we shall study the spectrality of $\mu _{M,D}$ in the case when $|\det (M)|=p$ is a prime. The main result shows that under certain mild conditions, if there are two points $s_{1}, s_{2}\in \mathbb R ^{n}, s_{1}-s_{2}\in \mathbb Z ^{n}$ such that the exponential functions $e_{s_{1}}(x), e_{s_{2}}(x)$ are orthogonal in $L^{2}(\mu _{M,D})$ , then the self-affine measure $\mu _{M,D}$ is a spectral measure with lattice spectrum. This gives some sufficient conditions for a self-affine measure to be a lattice spectral measure.     

The probability distribution and Hausdorff dimension of self-affine functions

Summary A simple natural measure is found with respect to which the probability distribution of a continuous self-affine functionf in the sense of Kôno is absolutely continuous. As an immediate corollary we obtain the result of Kôno that provides a necessary and sufficient condition for this distribution to be absolutely continuous with respect to Lebesgue measure. For the class of continuous self-affine functions one proves the conjecture of T. Bedford which says in this context that the Hausdorff dimension of the graph off is equal to its box dimension if and only if the probability distribution off is absolutely continuous with respect to Lebesgue measure.     

Integral self-affine sets with positive Lebesgue measures

A self-affine region is an integral self-affine set with positive Lebesgue measure. In this note we give two criteria for integral self-affine sets being self-affine regions. As their applications we study the L ¹-solutions of refinement equations, which play an important role in constructing wavelets, and we give several interesting examples. Received: 8 May 2007     

Analysis of μ R,D -Orthogonality in Affine Iterated Function Systems

Let μ _R,D be a self-affine measure associated with an expanding integer matrix R∈M _n (ℤ) and a finite subset D⊆ℤ ⁿ . In the present paper we study the μ _R,D -orthogonality and compatible pair conditions. We also show that any set of μ _R,D -orthogonal exponentials contains at most 3 elements on the generalized plane Sierpinski gasket and the number 3 is the best.      

Fuzzy measure of fuzzy events defined by fuzzy integrals

We define the concept of fuzzy measure of a fuzzy event by using a general form of fuzzy integral proposed by Murofushi, called fuzzy t-conorm integral, encompassing previous definitions. Zadeh defined the probability measure of a fuzzy event, and later the possibility measure of fuzzy event. Using a duality property of fuzzy t-conorm integral, we propose a general definition of fuzzy measure of fuzzy events, which is compatible with previous definitions of Zadeh, and possesses all properties of a fuzzy measure, in particular the duality property. Using our definition, we examine the case of decomposable measures and belief functions. A comparison with previous works is provided.     

Non-differentiability and Hölder properties of self-affine functions

We consider the class of self-affine functions. Firstly, we characterize all nowhere differentiable self-affine continuous functions. Secondly, given a self-affine continuous function $?$, we investigate its Hölder properties. We find its best uniform Hölder exponent and when $?$ is $C^1$, we find the best uniform Hölder exponent of ${?}^{\prime }$. Thirdly, we show that the Hölder cut of $?$ takes the same value almost everywhere for the Lebesgue measure. This last result is a consequence of the Borel strong law of large numbers.