齐次集及其拟Lipschitz等价

本文定义并研究一类齐次分形,该类分形包含所有的(拟)Ahlfors-David正则集和许多非正则的Moran集,这里如果一个分形的Hausdorff维数与packing维数不相等,则称它是非正则的.对于这类齐次分形,本文得出它们的分形维数,并且给出在适当分离条件下两个齐次分形拟Lipschitz等价的充要条件.随后,本文将这些结果应用到非正则的Moran集上.     

相应于随机自相似分形的记忆函数和分数次积分

For a physics system which exhibits memory, if memory is preserved only at points of random self-similar fractals, we define random memory functions and give the connection between the expectation of flux and the fractional integral. In particular, when memory sets degenerate to Cantor type fractals or non-random self-similar fractals our results coincide with that of Nigmatullin and Ren et al. .     

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．     

Scaling of mathematical fractals and box-counting quasi-measure

The same term, ‘fractals’ incorporates two rather different meanings and it is convenient to split the term into physical or empirical fractals and mathematical ones. The former term is used when one considers real world or numerically simulated objects exhibiting a particular kind of scaling that is the so-called fractal behaviour, in a bounded range of scales between upper and lower cutoffs. The latter term means sets having non-integer fractal dimensions. Mathematical fractals are often used as models for physical fractal objects. Scaling of mathematical fractals is considered using the Barenblatt–Borodich approach that refers physical quantities to a unit of the fractal measure of the set. To give a rigorous treatment of the fractal measure notion and to develop the approach, the concepts of upper and lower box-counting quasi-measures are presented. Scaling properties of the quasi-measures are studied. As examples of possible applications of the approach, scaling properties of the problems of fractal cracking and adsorption of various substances to fractal rough surfaces are discussed.     

Existence of self-similar energies on finitely ramified fractals

A self-similar energy on finitely ramified fractals can be constructed starting from an eigenform, i.e., an eigenvector of a special operator defined on the fractal. In this paper, we prove two existence results for regular eigenforms that consequently are existence results for self-similar energies on finitely ramified fractals. The first result proves the existence of a regular eigenform for suitable weights on fractals, assuming only that the boundary cells are separated and the union of the interior cells is connected. This result improves previous results and applies to many finitely ramified fractals usually considered. The second result proves the existence of a regular eigenform in the general case of finitely ramified fractals in a setting similar to that of P.C.F. self-similar sets considered, for example, by R. Strichartz in [11]. In this general case, however, the eigenform is not necessarily on the given structure, but is rather on only a suitable power of it. Nevertheless, as the fractal generated is the same as the original fractal, the result provides a regular self-similar energy on the given fractal.     

Dimension of Slices Through Fractals with Initial Cubic Pattern

In this paper, the Hausdorff dimension of the intersection of self-similar fractals in Euclidean space R~n generated from an initial cube pattern with an(n-m)-dimensional hyperplane V in a fixed direction is discussed. The authors give a sufficient condition which ensures that the Hausdorff dimensions of the slices of the fractal sets generated by "multirules" take the value in Marstrand's theorem, i.e., the dimension of the self-similar sets minus one. For the self-similar fractals generated with initial cube pattern, this sufficient condition also ensures that the projection measure μVis absolutely continuous with respect to the Lebesgue measure L~m. When μV《 L~m, the connection of the local dimension ofμVand the box dimension of slices is given.     

随机分形的测度性质

本文主要介绍随机过程样本轨道、Ｈａｗｋｅｓ模型、统计自相似集、统计自仿射集的测度性质，同时也将介绍一些离散分形的结果．文中还列出一些尚未解决的问题．     

直线上分形的Hausdorff测度的一个算法

本文给出了直线上Marion集的Hausdorff测度的一个有效计算方法,并通过几个实例得出如何利用此方法计算出直线上分形的Hausdorff测度的精确值.     

Discrete characterisations of Lipschitz spaces on fractals

A. Kamont has discretely characterised Besov spaces on intervals. In this paper, we give a discrete characterisation of Lipschitz spaces on fractals admitting a type of regular sequence of triangulations, and for a class of post critically finite self‐similar sets. This shows that on some fractals, certain discretely defined Besov spaces, introduced by R. Strichartz, coincide with Lipschitz spaces introduced by A. Jonsson and H. Wallin for low order of smoothness. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Scaling of mathematical fractals and box-counting quasi-measure

The same term, ‘fractals’ incorporates two rather different meanings and it is convenient to split the term into physical or empirical fractals and mathematical ones. The former term is used when one considers real world or numerically simulated objects exhibiting a particular kind of scaling that is the so-called fractal behaviour, in a bounded range of scales between upper and lower cutoffs. The latter term means sets having non-integer fractal dimensions. Mathematical fractals are often used as models for physical fractal objects. Scaling of mathematical fractals is considered using the Barenblatt–Borodich approach that refers physical quantities to a unit of the fractal measure of the set. To give a rigorous treatment of the fractal measure notion and to develop the approach, the concepts of upper and lower box-counting quasi-measures are presented. Scaling properties of the quasi-measures are studied. As examples of possible applications of the approach, scaling properties of the problems of fractal cracking and adsorption of various substances to fractal rough surfaces are discussed.     

On the connectedness of limit net sets

In this paper we introduce and study net sets and limit net sets. The construction and geometry of net sets can be described with the help of substitutions with net matrices which we also introduce here. Limit net sets are a special type of Moran fractals. We study connectedness properties of net sets and limit net sets.     

Singular sets of Lebesgue integrable functions

We discuss the relation of Lebesgue integrability of some functions generated by fractal sets to Minkowski contents and box dimensions of fractals. A Lebesgue integrable function is constructed which is maximally singular in the sense that the Hausdorff dimension of its singular set is equal to N.     

Classification of Moran fractals

In the paper, we try to classify Moran fractals by using the quasi-Lipschitz equivalence, and prove that two regular homogeneous Moran sets are quasi-Lipschitz equivalent if and only if they have the same Hausdorff dimension.     

Some fractals associated with Cantor expansions

In this paper, we consider a class of fractals generated by the Cantor series expansions. By constructing some homogeneous Moran subsets, we prove that these sets have full dimension.     

MULTIFRACTAL DECOMPOSITION OF CERTAIN RECURSIVE SETS WITHOUT THE OPEN SET CONDITION

1 IntroductionMultifractal decomposition has been investigated by rnany authors, it has become a use-ful tool in the fractal allalysis. Cawley and Mauldin had given good results on Moran fractaldecomPosition[3], Edgar and Mauldin studied the degraph multifractals[4], Falconer random-nized Cawley's results and the latest results on random selfsidrilar multifractals were doneby Arbeiter and Patzschke under rather weak .o1ldition.I2J. However, all these results wereestablished on certain sepa…     

Elementary density bounds for self-similar sets and application

Falconer[1] used the relationship between upper convex density and upper spherical density to obtain elementary density bounds for s-sets at H S-almost all points of the sets. In this paper, following Falconer[1], we first provide a basic method to estimate the lower bounds of these two classes of set densities for the self-similar s-sets satisfying the open set condition （OSC）, and then obtain elementary density bounds for such fractals at all of their points. In addition, we apply the main results to the famous classical fractals and get some new density bounds.     

Completeness of Muckenhoupt classes

In this note we prove that the Hausdorff distance between compact sets and the Kantorovich distance between measures, provide an adequate setting for the convergence of Muckenhoupt weights. The results which we prove on compact metric spaces with finite metric dimension can be applied to classical fractals.     

Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups

We solve Gromov's dimension comparison problem for Hausdorff and box counting dimension on Carnot groups equipped with a Carnot-Carathéodory metric and an adapted Euclidean metric. The proofs use sharp covering theorems relating optimal mutual coverings of Euclidean and Carnot-Carathéodory balls, and elements of sub-Riemannian fractal geometry associated to horizontal self-similar iterated function systems on Carnot groups. Inspired by Falconer's work on almost sure dimensions of Euclidean self-affine fractals we show that Carnot-Carathéodory self-similar fractals are almost surely horizontal. As a consequence we obtain explicit dimension formulae for invariant sets of Euclidean iterated function systems of polynomial type. Jet space Carnot groups provide a rich source of examples.     

Fractal properties of refined box dimension on functional graph

Many fractals, which have theoretical and practical significance, take the form of functional graphs. For continuous functions whose graphs are fractal sets, their fractal characteristics are studied, and the relations between refined box dimensions of functional graphs before and after four arithmetic operations are discussed.     

Dimension matrix of the G-M fractal

Fractals which represent many of the sets in various scientific fields as well as in nature is geometrically too complicate. Then we usually use Hausdorff dimension to estimate their geometrical properties. But to explain the fractals from the Hausdorff dimension induced by the Euclidan metric are not too sufficient. For example, in digital communication, while encoding or decoding the fractal images, we must consider not only their geometric sizes but also many other factors such as colours, densities and energies etc.. So in this paper we define the dimension matrix of the sets by redefining the new metric.