The Multifractal Formalism for Measures,Review and Extension to Mixed Cases

The multifractal formalism for single measure is reviewed.Next,a mixed generalized multifractal formalism is introduced which extends the multifractal formalism of a single measure based on generalizations of the Hausdorff and packing measures to a vector of simultaneously many measures.Borel-Cantelli and Large deviations Theorems are extended to higher orders and thus applied for the validity of the new variant of the multifractal formalism for some special cases of multi-doubling type measures.     

Multifractal Components of Multiplicative Set Functions

We analyze the multifractal spectrumof multiplicative set functions on a self-similar set with open set condition. We show that the multifractal components carry self-similar measures which maximize the dimension. This gives the dimension of a multifractal component as the solution of a problem of maximization of a quasiconcave function satisfying a set of linear constraints. Our analysis covers the case of multifractal components of self-similar measures, the case of Besicovitch normal sets of points, the multifractal spectrum of the relative logarithmic density of a pair of self-similar measures, the multifractal spectrum of the Liapunov exponent of the shift mapping and the intersections of all these sets. We show that the dimension of an arbitrary union of multifractal components is the supremum of the dimensions of the multifractal components in the union. The multidimensional Legendre transform is introduced to obtain the dimension of the intersection of finitely many multifractal components.     

Dynamical multifractal zeta-functions,multifractal pressure and fine multifractal spectra

We introduce multifractal pressure and dynamical multifractal zeta-functions, providing precise information on a very general class of multifractal spectra, including, for example, the fine multifractal spectra of self-conformal measures and the fine multifractal spectra of ergodic Birkhoff averages of continuous functions.     

Multivariate multifractal analysis

We show how a joint multifractal analysis of a collection of signals unravels correlations between the locations of their pointwise singularities. The multivariate multifractal formalism, reformulated in the general setting supplied by multiresolution quantities, provides a framework which allows to estimate joint multifractal spectra. General results on joint multifractal spectra are derived, and illustrated by the theoretical derivation and practical estimation of the joint multifractal spectra of simple mathematical models, including correlated binomial cascades.     

A multifractal formalism in a probability space

In this paper, we mainly research a multifractal formalism in a probability space.The first objective of this paper is to define a multifractal generalisation of the Hausdorff measure and the packing measure. We explore the properties of the multifractal Hausdorff measure and the multifractal packing measure in a probability space, and invesigate the relative results of multifractal spectra functions in a probability space. We also describe a sufficient condion and a necessary one of validity for the multifractal formalism in a probability space.     

Multifractal analysis of a measure of multifractal exact dimension

We focus on the multifractal generalization of the centered Hausdorff measure and dimension. We analyze the correlation among different approaches to the definition of the multifractal exact dimension of locally finite and Borel regular measures on the basis of fractal analysis of essential supports of these measures. Using characteristic multifractal measures, we carry out the multifractal analysis of singular probability measures and prove theorems on the structural representation of these measures.     

On the Projections of the Mutual Multifractal Rényi Dimensions

In this paper, we compare the mutual multifractal Rényi dimensions to the mutual multifractal Hausdorff and pre-packing dimensions. We also provide a relationship between the mutual multifractal Rényi dimensions of orthogonal projections of a couple of measures $(\mu,\nu)$ in $\mathbb{R}^n$. As an application, we study the mutual multifractal analysis of the projections of measures.     

Variational principles and mixed multifractal spectra

We establish a ``conditional' variational principle, which unifies and extends many results in the multifractal analysis of dynamical systems. Namely, instead of considering several quantities of local nature and studying separately their multifractal spectra we develop a unified approach which allows us to obtain all spectra from a new multifractal spectrum. Using the variational principle we are able to study the regularity of the spectra and the full dimensionality of their irregular sets for several classes of dynamical systems, including the class of maps with upper semi-continuous metric entropy.

Another application of the variational principle is the following. The multifractal analysis of dynamical systems studies multifractal spectra such as the dimension spectrum for pointwise dimensions and the entropy spectrum for local entropies. It has been a standing open problem to effect a similar study for the ``mixed' multifractal spectra, such as the dimension spectrum for local entropies and the entropy spectrum for pointwise dimensions. We show that they are analytic for several classes of hyperbolic maps. We also show that these spectra are not necessarily convex, in strong contrast with the ``non-mixed' multifractal spectra.

     

北京交通流序列的重分形扩散熵分析

运用重分形扩散熵分析方法来分析北京交通拥堵指数的长程相关性和重分形特征.方法综合使用了扩散技术和Renyi熵来研究北京交通拥堵指数的标度行为.由于交通拥堵指数序列具有明显的周期性,故先选用傅里叶滤波去除序列的周期性,再进行重分形扩散熵分析.实验结果表明北京交通拥堵指数序列的极端波动显示出反相关性,同时拥堵指数序列具有较弱的重分形特征.     

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…     

The multifractal structure of the product of two stable occupation measures

We study the multifractal structure of the product of two stable occupation measures and obtain the multifractal spectrum.     

Multifractal spectra and multifractal zeta-functions

We introduce multifractal zetafunctions providing precise information of a very general class of multifractal spectra, including, for example, the multifra     

一类自相似集重分形分支的量纲

本文首先定义具有量纲函数的重分形测度,然后证明当Euclid空间中的两个重分形测度具有等价的量纲函数时,它们也等价.进一步,对于直线上满足强分离条件(SSC)的自相似集,在某些加倍条件下,本文给出判断其重分形分支的量纲函数的充要条件.     

Multifractal nature of extensive air showers

Cherenkov images are multifractal in nature. We show that multifractal behaviour of Cherenkov images arises due to multiplicative nature of pair production and bremsstrahlung processes in the longitudinal shower development passage.     

求多重分形谱的滑动格子法在深浅地表元素分布中的应用研究

在多重分形理论和特征判定法的基础上,构造了求多重分形谱的滑动格子计算法,计算出了研究区域4种元素深、浅层的多重分形谱f(α)的图像.结果显示浅层元素的分布不具备多重分形特征;深层元素分布符合多重分形特征.就三种分形维数——格子维数、信息维数、关联维数对深层元素的分布做出了大小排序解释;后就多重分形谱f(α)的跨度、对称性和两端差值Δf做出了对应于深层元素分布概率分布集中差异、高低浓度分布差异、稳定性的解释.最后根据上述分析的结果指出应用求多重分形谱的滑动格子法研究深浅地层元素分布是一快速、实用、有效的方法,具有良好的应用前景.     

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…     

Multifractal analysis of local entropies for recurrence time

We introduce local entropies and multifractal spectra associated with Poincaré recurrences. By using characteristics of a dynamical system we establish an exact formula on multifractal spectrum of local entropies for recurrence time.     

From Multifractal Measures to Multifractal Wavelet Series

Given a positive locally finite Borel measure μ on R, a natural way to construct multifractal wavelet series is to set , where . Indeed, under suitable conditions, it is shown that the function F_μ inherits the multifractal properties of μ. The transposition of multifractal properties works with many classes of statistically selfsimilar multifractal measures, enlarging the class of processes which have self-similarity properties and controlled multifractal behaviors. Several perturbations of the wavelet coefficients and their impact on the multifractal nature of F_μ are studied. As an application, multifractal Gaussian processes associated with F_μ are created. We obtain results for the multifractal spectrum of the so-called W-cascades introduced by Arnéodo et al.