THE MULTIFRACTAL HAUSDORFF AND PACKING MEASURE OF GENERAL SIERPINSKI CARPETS

1 IntroductionThe self-affine sets include self-similar sets as their special case. Although the fractalproperties of self-similar sets are well understood, little is known about self-affine sets in general.McMullen[1] studied a class of self~affine sets called generlized Sierpinski carpets, and got theirHausdorff and box dimensions. King[2] got the singular spectrum of general Sierpinski carpets.In [3] Olsen introduced the multifratal Hausdorff ajnd packing measure. and use them tostudy th…     

Continuous dependence on parameters of certain self-affine measures,and their singularity

In this paper, we first prove that the self-affine sets depend continuously on the expanding matrix and the digit set, and the corresponding self-affine measures with respect to the probability weight behave in much the same way. Moreover, we obtain some sufficient conditions for certain self-affine measures to be singular.     

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.     

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.     

Spectra of a class of self-affine measures

In the present paper we will study the spectral property of a class of self-affine measures under the condition of compatible pair. We first answer a question of Dutkay and Jorgensen concerning the relationship between spectral self-affine measure and compatible pair. We then consider the spectra of Bernoulli convolutions and obtain a sharp result which extends the corresponding result of Jorgensen, Kornelson and Shuman. Finally, we provide a structural property for the integer spectrum of a spectral self-affine measure.     

Fractional differentiation in the self-affine case. IV — random measures

The concept of fractional differentiation of stochastic processes introduced in Part I is translated to fractional densities of random measures. Thereby we consider self-affine random measures in the constructive and the axiomatic sense. Occupation measures and local time measures of self-affine random fields are special examples.     

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.     

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-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     

Non-conformal Repellers and the Continuity of Pressure for Matrix Cocycles

The pressure function P(A, s) plays a fundamental role in the calculation of the dimension of “typical” self-affine sets, where A = (A ₁, …,A _k ) is the family of linear mappings in the corresponding generating iterated function system. We prove that this function depends continuously on A. As a consequence, we show that the dimension of “typical” self-affine sets is a continuous function of the defining maps. This resolves a folklore open problem in the community of fractal geometry. Furthermore we extend the continuity result to more general sub-additive pressure functions generated by the norm of matrix products or generalized singular value functions for matrix cocycles, and obtain applications on the continuity of equilibrium measures and the Lyapunov spectrum of matrix cocycles.     

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.     

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.     

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.     

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.     

Invariant measures for non-primitive tiling substitutions

We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well known that in the primitive case, the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution is not primitive and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some mild assumptions, we completely characterize σ-finite invariant measures which are positive and finite on a cylinder set. A key step is to establish recognizability of non-periodic tilings in our setting. Examples include the “integer Sierpiński gasket and carpet” tilings. For such tilings, the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported σ-finite invariant measure that is locally finite and unique up to scaling.     

On a family of singular continuous measures related to the doubling map

Here, we study some measures that can be represented by infinite Riesz products of 1-periodic functions and are related to the doubling map. We show that these measures are purely singular continuous with respect to Lebesgue measure and that their distribution functions satisfy super-polynomial asymptotics near the origin, thus providing a family of extremal examples of singular measures, including the Thue–Morse 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.     

A sharp bound on the Hausdorff dimension of the singular set of a uniform measure

The study of the geometry of n-uniform measures in \(\mathbb {R}^{d}\) has been an important question in many fields of analysis since Preiss’ seminal proof of the rectifiability of measures with positive and finite density. The classification of uniform measures remains an open question to this day. In fact there is only one known example of a non-trivial uniform measure, namely 3-Hausdorff measure restricted to the Kowalski–Preiss cone. Using this cone one can construct an n-uniform measure whose singular set has Hausdorff dimension \(n-3\). In this paper, we prove that this is the largest the singular set can be. Namely, the Hausdorff dimension of the singular set of any n-uniform measure is at most \(n-3\).     

Singular time changes of diffusions on Sierpinski carpets

In this study we construct self-similar diffusions on the Sierpinski carpet that are reversible with respect to the Hausdorff measure. The diffusions are obtained from self-similar diffusions reversible with respect to self-similar measures, which are singular to the Hausdorff measure. To do this we introduce a new sufficient condition for the continuity of sample paths to be preserved by a singular time change.     

支于自仿射Tile上的细分分布

In this paper,some conditions which assure the compactly supported refinable distributions supported on a self-affine tile to be Lebesgue-Stieltjes measures or absolutely continuous measures with respect to Lebesgue-Stieltjes measures are given.