自相似测度的联合发散点和填充维数

近年来,发散点引起了数学界的广泛关注,对单测度的发散点,前人已经作了很完备的研究,对有限多个自相似测度的联合发散点的集合,仅其豪斯多夫维数被L．Olsen研究并给出了,然而,我们对其填充维数仍一无所知,因此,本文主要研究联合发散点集合的填充维数。     

Cookie-Cutter~集上的~Gibbs~测度

Cookie-Cutter集不仅是动力系统中重要的研究对象,而且是分形中一类重要的集合.而支撑在其上的Gibbs测度对计算分形维数和热力学机制的熵起关键性作用.本文借助于定理2.1构造了支撑在其上的Gibbs测度,并用遍历性证明了该测度的唯一性.     

类切饼集的测度维数

本文考虑了满足一定条件的类切饼集,主要利用密度定理,获得其Hausdorff测度和填充测度是等价的.同时,给出这类类切饼集维数和测度的统计解释.     

Cantor集随机重排的填充测度

获得了Cantor集随机重排后所得的随机集的填充测度,还得到了一般随机集的填充维数及某些“正则”序列所产生的随机集的Hausdorff测度及填充测度。     

单峰映射允许揉搓序列的维数和测度

本文证明了单峰映射的允许揉搓序列组成的集合在符号空间∑2中的Hausdorff维数为1,1维Hausdorff测度为零.     

填充维数的一个等价定义

本文研究了填充维数与上盒维数的关系.利用Cantor-Bendixson定理的方法,得到了由上盒维数给出的填充维数的等价定义.并证明了齐次Moran集对上盒维数和填充维数的连续性.     

R~n上分形集的多重维数

本文推广Ｈａｕｓｄｏｒｆｆ测度和维数的概念，引入了被称作为多重维测度和多重维数的概念．文中证明了关于多重维测度的Ｆｒｏｓｔｍａｎ定理，构造了一个例子说明存在一类点集，其Ｈａｕｓｄｏｒｆｆ测度是零或十∞，但其多重维测度是一个正数，并说明了多重维数除第一个分量是正数外，其它分量可以取到任何实数．     

关于测度的Hausdorff维数的局部化

主要研究测度的豪斯道夫维数的局部化.通过定义一个测度μx,ε,从而给出dim·Hμ在点x的局部化维数dim·Hμ(x).进而得到局部化维数dim·Hμ(x)与dim·Hμ之间的关系,并得到了一个等式关系.     

Z~d-作用下的Bowen拓扑熵理论

在Z~d-作用下定义了Bowen维数熵,研究了Z~d-作用下Bowen维数熵的一些性质,证明了X中的任意子集的Bowen维数熵可以通过该子集中的点的测度下局部熵估计.     

双曲空间上超布朗运动的范围

对双曲空间上的超布朗运动进行了研究,证明了该过程的范围属于某集合的概率可以表示为一个奇异边值问题的解.在得到了这个解的一个极限结果以后,给出了上述概率的一个渐近行为.对于维数d≥2,还证明了从黎曼测度出发的超过程在任何内点非空的有界Borel子集上的占位时是无穷的结论.     

Divergence points of self-similar measures and packing dimension

Let μ be a self-similar measure in Rd. A point x∈Rd for which the limit does not exist is called a divergence point. Very recently there has been an enormous interest in investigating the fractal structure of various sets of divergence points. However, all previous work has focused exclusively on the study of the Hausdorff dimension of sets of divergence points and nothing is known about the packing dimension of sets of divergence points. In this paper we will give a systematic and detailed account of the problem of determining the packing dimensions of sets of divergence points of self-similar measures. An interesting and surprising consequence of our results is that, except for certain trivial cases, many natural sets of divergence points have distinct Hausdorff and packing dimensions.     

Hausdorff and packing dimensions of non-normal tuples of numbers: Non-linearity and divergence points

We apply the results in [L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, J. Math. Pures Appl. 82 (2003) 1591-1649; L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. III, Aequationes Math. 71 (2006) 29-53; L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. IV: Divergence points and packing dimension, Bull. Sci. Math. 132 (2008) 650-678; L. Olsen, S. Winter, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. II: Non-linearity, divergence points and Banach space valued spectra, Bull. Sci. Math. 131 (2007) 518-558] to give a systematic and detailed account of the Hausdorff and packing dimensions of sets of d-tuples of numbers defined in terms of the asymptotic behaviour of the frequencies of strings of digits in their N-adic expansion.     

The dimensions of the divergence points of self-similar measures with weak separation condition

Let \(\mu \) be the self-similar measure supported on the self-similar set K with the weak separation condition, which is weaker than the open set condition. This article uses Hausdorff dimension and packing dimension to investigate the multifractal structure of several sets of divergence points of \(\mu \) in the iterated function system.     

Applications of multifractal divergence points to some sets of d-tuples of numbers defined by their N-adic expansion

In this paper we apply the techniques and results from the theory of multifractal divergence points developed in [L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, Journal de Mathématiques Pures et Appliquées 82 (2003) 1591-1649; L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages III, Preprint (2002); L. Olsen, S. Winter, J. London Math. Soc. 67 (2003) 103-122; L. Olsen, S. Winter, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages II, Preprint (2001)] to give a systematic and detailed account of the Hausdorff dimensions of sets of d-tuples numbers defined in terms of the asymptotic behaviour of the frequencies of the digits in their N-adic expansion. Using the method and results from [L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, Journal de Mathématiques Pures et Appliquées 82 (2003) 1591-1649; L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages III, Preprint (2002); L. Olsen, S. Winter, J. London Math. Soc. 67 (2003) 103-122; L. Olsen, S. Winter, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages II, Preprint (2001)] we investigate and compute the Hausdorff dimension of several new sets of d-tuples of numbers. In particular, we compute the Hausdorff dimension of a large class of sets of d-tuples numbers for which the limiting frequencies of the digits in their N-adic expansion do not exist. Such sets have only very rarely been studied. In addition, our techniques provide simple proofs of higher-dimensional and non-linear generalizations of known results, by Cajar and Volkmann and others, on the Hausdorff dimension of sets of normal and non-normal numbers.     

Moduli of continuity of a class of N-parameter Gaussian processes and their fast points

We study the moduli of continuity of a class of N-parameter Gaussian processes and get some results on＇the packing dimension of the set of their fast points.     

Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. IV: Divergence points and packing dimension

During the past 10 years multifractal analysis has received an enormous interest. For a sequence n(φn) of functions on a metric space X, multifractal analysis refers to the study of the Hausdorff and/or packing dimension of the level sets(1) of the limit function limnφn. However, recently a more general notion of multifractal analysis, focusing not only on points x for which the limit limnφn(x) exists, has emerged and attracted considerable interest. Namely, for a sequence n(xn) in a metric space X, we let A(xn) denote the set of accumulation points of the sequence n(xn). The problem of computing that the Hausdorff dimension of the set of points x for which the set of accumulation points of the sequence (φnn(x)) equals a given set C, i.e. computing the Hausdorff dimension of the set(2){x∈X|A(φn(x))=C} has recently attracted considerable interest and a number of interesting results have been obtained. However, almost nothing is known about the packing dimension of sets of this type except for a few special cases investigated in [I.S. Baek, L. Olsen, N. Snigireva, Divergence points of self-similar measures and packing dimension, Adv. Math. 214 (2007) 267–287]. The purpose of this paper is to compute the packing dimension of those sets for a very general class of maps φn, including many examples that have been studied previously, cf. Theorem 3.1 and Corollary 3.2. Surprisingly, in many cases, the packing dimension and the Hausdorff dimension of the sets in (2) do not coincide. This is in sharp contrast to well-known results in multifractal analysis saying that the Hausdorff and packing dimensions of the sets in (1) coincide.     

Effective fractal dimensions

Classical fractal dimensions (Hausdorff dimension and packing dimension) have recently been effectivized by (i) characterizing them in terms of real‐valued functions called gales, and (ii) imposing computability and complexity constraints on these gales. This paper surveys these developments and their applications in algorithmic information theory and computational complexity theory. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Divergence points of self-similar measures satisfying the OSC

Recently, Barreira and Schmeling (2000) [1] and Chen and Xiong (1999) [2] have shown, that for self-similar measures satisfying the SSC the set of divergence points typically has the same Hausdorff dimension as the support K. It is natural to ask whether we obtain a similar result for self-similar measures satisfying the OSC. However, with only the OSC satisfied, we cannot do most of the work on a symbolic space and then transfer the results to the subsets of Rd, which makes things more difficult. In this paper, by the box-counting principle we show that the set of divergence points has still the same Hausdorff dimension as the support K for self-similar measures satisfying the OSC.     

Dimensions of subsets of Moran fractals related to frequencies of their codings

The Moran fractal considered in this paper is an extension of the self-similar sets satisfying the open set condition. We consider those subsets of the Moran fractal that are the union of an uncountable number of sets each of which consists of the points with their location codes having prescribed mixed group frequencies. It is proved that the Hausdorff and packing dimensions of each of these subsets coincide and are equal to the supremum of the Hausdorff (or packing) dimensions of the sets in the union. An approach is given to calculate their Hausdorff and packing dimensions. The main advantage of our approach is that we treat these subsets in a unified manner. Another advantage of this approach is that the values of the Hausdorff and packing dimensions do not need to be guessed a priori.