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The refined multifractal formalism of some homogeneous Moran measures
Abstract
The concept of dimension is an important task in geometry. It permits a description of the growth process of objects. It may be seen as an invariant measure
characterizing the object. Fractal dimensions are a kind of invariants permitting essentially to describe the irregularity hidden in irregular objects, by providing a suitable growth law. Among fractal geometrical objects, Moran’s types play an important role in explaining many situations, in pure mathematics as the general context of Cantor’s, and in applied physics as a suitable context for studying scaling laws. In the present paper, some non-regular homogeneous Moran measures are investigated, by establishing some new sufficient conditions permitting an explicit computation of the relative multifractal dimensions of the level sets for which the classical formulation does not hold. Besides, the mutual singularity of the relative multifractal measures for the homogeneous Moran case with different multifractal dimensions is investigated. This is very important, as in quasi-all existing situations, the validity of the multifractal formalism passes through the equality of the multifractal Hausdorff dimension with the packing one.
... Numerous writers have looked at these measures, stressing their significance for the study of local fractal properties and fractal products [2,4,24,25,26,33,43]. In particular, Cole introduced in [14] a generalized Hausdorff and packing measures H q,s µ and P q,s µ respectively (see [1,2,4,6,17,19,34,48,50,51] for more details on these measures and their applications) in order to introduce a general formalism for the multifractal analysis of one probability measure µ with respect to an other measure ν. More specifically, he calculated, for α ≥ 0, the size of the set E(α) = x ∈ supp µ ∩ supp ν; lim r→0 log µ(B(x, r)) log ν(B(x, r)) = α , (1.1) where supp µ is the topological support of the measure µ and B(x, r) stand for the closed ball of center x and radius r. ...
... These are straightforward analogues of the so-called thermodynamic formalism's characterization of the dimensions of repellers. We define now the pressure function Π and the Gibbs measure π q,t (see [19]) respectively by ...
... A simple calculation gives that Given q ∈ R, there exists from [19] a probability measure π q,β(q) supported by E such that for any n ≥ 1 and σ 0 ∈ D n , π q,β(q) (σ 1 σ 2 · · · σ n ) = µ (I σ0 ) q ν (I σ0 ) β(q) σ∈Dn µ (I σ ) q ν (I σ ) β(q) . ...
On a Moran set meeting the strong separation requirement, we examine a family of multi-fractal Hausdorff and packing measures and dimensions in this study. Let ϕ * π be the image measure of ergodic Borel probability measure π and measurable function ϕ. Entropy is used to calculate the formula for the dimension of the multifractal measure of ϕ * π. Moreover, some statistical interpretations, on a class of Moran sets, of the dimensions and corresponding geometrical measures are also supported. Finally, a specific illustration of a measure that satisfies the aforementioned criteria is created.
... Before we set our main results, let us recall the class of homogeneous Moran sets (see the references 10,11,42 ). Let {n v } v≥1 be a sequence of positive integers and {c v } v≥1 be a sequence of positive numbers satisfying ...
... Definition 2. 10,11,42 Let I be a closed interval of length 1. The following conditions allows us to describe a Moran structure for which ...
There are only two kinds of measures in which the mixed multifractal formalism applies, which are self-similar and self-conformal measures. This article is about studying the validity and non-validity of the mixed multifractal formalism of other kinds of measures, called irregular/homogeneous Moran measures.
... In several recent papers, the mentioned type of multifractal analysis has re-emerged as mathematicians and physicists have begun to discuss the idea of performing multifractal analysis with respect to an arbitrary reference measure, see for example. [8][9][10][11][12][13][14][15][16][17][18] Throughout this paper, q, t ∈ R, B(x, r) stands for the closed ball ...
Sets of infinite multifractal measures are awkward to work with, and reducing them to sets of positive finite multifractal measures is a very useful simplification. The aim of this paper is to show that the multifractal Hausdorff measures satisfy the "subset of positive and finite measure" property. We apply our main result to prove that the multifractal function dimension is defined as the supremum over the multifractal dimension of all Borel probability measures.
... These sets may be understood as attractors for dynamical systems, electrical circuits, and also smart cities where fractals are nowadays sophisticated tools in their modeling. Fractals such as Cantor and Moran's types in general are also applied in understanding physical properties at different molecular levels, such as nonmaterial composites, crystal growth, and structure, porous materials, etc. [9]. Finally, one can note some investigations in multifractal analysis of Moran sets: multifractal properties of homogeneous Moran fractals associated with Fibonacci sequence [43], multifractal properties [42,44]. ...
This article is devoted to sets having the Moran structure. The main attention is given to topological, metric, and fractal properties of certain sets whose elements have restrictions on using digits or combinations of digits in own representations.
... Section 2 explores multifractal analysis on the Moran sets and its corresponding measures. For the relative multifractal formalism of non-regular Moran measures, some sufficient conditions are developed [22]. Further, the idea of a stochastic dyadic Cantor set is extended to a weighted planar stochastic lattice, which leads to a stochastic porous lattice [23]. ...
This special issue of the European Physical Journal Special Topics titled “Frontiers of Fractals for Complex Systems: Recent Advances and Future Challenges” is a collection of cutting-edge research proposing the application of fractal features to the dynamics of highly nonlinear complex systems.
... The dimension is therefore a quantitative measure of growth. In fact, for studying physical systems in classical physics/mathematics, we consider the linear space in which the variables governing the physical system live, which is entirely contained in 3-dimensional Euclidean as real physical space (see for example [10,16]). ...
The main aim of this paper is to study average Hewitt-Stromberg and box dimensions of typical compact metric spaces belonging to the Gromov-Hausdorff space equipped with the Gromov-Hausdorff metric.
In the present work, we are concerned with the estimation of some mixed variants of multifractal dimensions for a special class of measures characterized by a weak Ahlfors assumption applying mixed multifractal generalizations of Hausdorff and packing measures. Exact computation of such dimensions is shown to be valid for a class of Moran-type measures in some special cases.
This study provides a general multifractal formalism that overcomes the limitations of traditional multifractal formalism. The generic Hewitt-Stromberg measures are used to introduce and study a multifractal formalism. The generic Hewitt-Stromberg dimensions' upper and lower bounds are estimated, producing results even at places q where the upper and lower multifractal Hewitt-Stromberg dimension functions diverge.
In this paper, the equivalence of the multifractal centered Hausdorff measure and the multifractal packing measure is investigated. Furthermore, for the Moran sets satisfying the strong separation condition, the equivalence of the mutual multifractal Hausdorff and packing measures is discussed. A concrete example of fractal sets satisfying the above property is developed.
In this paper, we study the mutual multifractal Hausdorff dimension
and the packing dimension of level sets K(\alpha, \beta) for some
non-regular Moran measures satisfying the so-called Strong
Separation Condition. We obtain sufficient conditions for the
valid multifractal formalisms of such measures and discuss some
examples.
In this work, some density estimations associated to vector-valued quasi-Ahlfors measures are developed within the mixed multifractal analysis framework. The principle idea reposes on the fact that being quasi-Ahlfors is sufficient to conduct a mixed multifractal analysis for vector-valued measures. In this work, we introduced a multifractal density for finitely many measures, and showed that such density may be estimated well by means of the mixed multifractal measures. Such estimation induces an exact computation of multifractal spectrum of the vector-valued quasi-Ahlfors measure.
This appendix gives a lower bound of the Billingsley-Hausdorff dimension of a level set related to the Birkhoff average in the "non-compact" symbolic space N^N, defined by Gibbs measure.
We propose in this paper an example for which the multifractal Hausdorff and packing measures are mutually singular.
The aim of this article is to show that the multifractal Hausdorff and packing measures are mutually singular, which in particular provides an answer to Olsen's questions.
A measure without local dimension is a measure such that local dimension does not exist for any point in its support. In this paper, we construct such a class of Moran measures and study their lower and upper local dimensions. We show that the related ‘free energy’ function (Lq-spectrum) does not exist. Nevertheless, we can obtain the full Hausdroff and packing dimension spectra for level sets defined by lower and upper local dimensions. They can be viewed as a generalized multifractal formalism.
Given two probability measures µ and ν on R n. We define the upper and lower relative multifractal box-dimensions of the measure µ with respect to the measure ν and investigate the relationship between the multifractal box-dimensions and the relative multifractal Hausdorff dimension, the relative multifractal pre-packing dimension. We also, calculate the relative multifractal spectrum and establish the validity of multifractal formalism. As an application, we study the behavior of projections of measures obeying to the relative multifractal formalism.
We model quantum transport, described by continuous-time quantum walks
(CTQW), on deterministic Sierpinski fractals, differentiating between
Sierpinski gaskets and Sierpinski carpets, along with their dual structures.
The transport efficiencies are defined in terms of the exact and the average
return probabilities, as well as by the mean survival probability when
absorbing traps are present. In the case of gaskets, localization can be
identified already for small networks (generations). For carpets, our numerical
results indicate a trend towards localization, but only for relatively large
structures. The comparison of gaskets and carpets further implies that,
distinct from the corresponding classical continuous-time random walk, the
spectral dimension does not fully determine the evolution of the CTQW.
Based on fractal properties and spatial auto-correlation, the
measures of complexity lacunarity, Moran's I and Geary's C index
are defined for 3D image analysis. Their abilities to investigate
translational invariance, characteristic length scales, spatial
correlation and shapes of 3D micro-structures are demonstrated on
proto-typical examples. Finally, using these measures of complexity,
3D images of trabecular bone are analysed. The main findings are
that the complexity of the trabecular structure decreases and the
plate-like shapes of the trabeculae change to mainly rod-like shapes
during bone loss. These results and the proposed measures could have
a great impact for medicine and for space exploration.
The aim of this work is the analysis of multifractals in the context of Mohamed El Naschie’s ϵ(∞) Cantorian space–time applied to cosmology. As starting point we consider the results of the first author of the present paper describing scaling rules in nature, R(N)=(h/mnc)Nϕ. Then, we use multifractal analysis to show that the result, already developed by the authors as Brownian motion, is a Multifractal process. Indeed, Brownian paths play a crucial role if considered to be a multifractal. Moreover, we summarize some recent results concerning fractal structure and the Brownian paths in order to calculate fractal dimension and characteristic parameters for large scale structures and for the atomic elements that live in an El Naschie’s ϵ(∞) Cantorian space–time.
The multifractal formalism for measures in its original formulation is checked for special classes of measures, such as, doubling, self-similar, and Gibbs-like ones. Out of these classes, suitable conditions should be taken into account to prove the validity of the multifractal formalism. In this work, a large class of measures satisfying a weak condition known as quasi-Ahlfors is considered in the framework of mixed multifractal analysis. A joint multifractal analysis of finitely many quasi-Ahlfors probability measures is developed. Mixed variants of multifractal generalizations of Hausdorff, and packing measures, and corresponding dimensions are introduced. By applying convexity arguments, some properties of these measures, and dimensions are established. Finally, an associated multifractal formalism is introduced, and proved to hold for the class of quasi-Ahlfors measures. Besides, some eventual applications, and motivations, especially, in AI are discussed.
We study the multifractal dimension functions of the Moran measure associated with a homogeneous (non-regular) Moran fractal and show that the multifractal measures are mutually singular when the multifractal Hausdorff and packing functions differ. As an application, we give concrete examples related to these concepts.
The aim of this article is to study the behavior of the multifractal packing function under slices in Euclidean space. We discuss the relationship between the multifractal packing and pre-packing functions of a compactly supported Borel probability measure and those of slices or sections of the measure. More specifically, we prove that if μ satisfies a certain technical condition and q lies in a certain somewhat restricted interval, then Olsen's multifractal dimensions satisfy the expected adding of co-dimensions formula.
Interpolation theory focuses on the existence of continuous functions which reconstruct the prescribed data set thereby helping to approximate the real-valued continuous function. In the historical development of the approximation theory, beyond polynomials, splines and trigonometric functions are likewise applied in approximation methods which are directed toward the smooth approximants. Then again, numerous practical and natural phenomenal signals reveal the non-smoothness in their traces and henceforth request non-smooth functions for significant reconstruction. The notion of a fractal function is explored based on the theory of iterated function system which was made by envisaging the universe as a fractal. Consequently, fractal functions are used for both smooth and non-smooth approximation by including various classical approximation methods, although there are differences between the fractal interpolation function and the traditional interpolation function (see for more details, [42–59]). Fractal interpolants are constructed by the theory of iterated function system which offers a self-referential functional equation for the interpolant and implies a self-similarity in magnification. Additionally, the choice of vertical scaling factors provides a flexible and optimal interpolant which also generates a specific traditional interpolant [47]. As a consequence of its fruitful success in non-smooth approximation, the theory of fractal functions has been extensively investigated, and there are developments in the pipeline beyond its mathematical framework. This chapter presents the construction of fractal interpolation functions and their evolution based on the vertical scaling factors and different iterated function systems.
Let ( X , d ) (X,d) be a metric space. For a probability measure μ \mu on a subset E E of X X and a Vitali cover V V of E E , we introduce the notion of a b μ b_{\mu } -Vitali subcover V μ V_{\mu } , and compare the Hausdorff measures of E E with respect to these two collections. As an application, we consider graph directed self-similar measures μ \mu and ν \nu in R d \mathbb {R}^{d} satisfying the open set condition. Using the notion of pointwise local dimension of μ \mu with respect to ν \nu , we show how the Hausdorff dimension of some general multifractal sets may be computed using an appropriate stochastic process. As another application, we show that Olsen’s multifractal Hausdorff measures are mutually singular.
In the present paper, we suggest new proofs of many known results about the relative multifractal formalism. We provide results even at points q for which the relative multifractal Hausdorff and packing functions differ. We also give some examples of two measures where the multifractal functions are different and the Hausdorff dimension of the level sets of the ν-local Hölder exponent is given by the Legendre transform of the multifractal Hausdorff function and their packing dimension by the Legendre transform of the multifractal packing function.
We study multifractal properties of Bernoulli measure μp supported on homogeneous Cantor set determined by ([0,1],(2)k≥1,(ck)k≥1), where 0<p<1/2 and ck is either d1 or d2 with 0<d1<d2<1/2. Let θ_ and θ‾ be the lower and upper limits of the occurrence frequency of d1 in (ck)k≥1, respectively. It is proved by Wu (Sci. China Ser. A 48, 2005) that, if θ_=θ‾ then the multifractal formalism holds. After given some conditions on (ck)k≥1, we get multifractal spectra for any 0≤θ_<θ‾≤1. We show that if θ_=0 and θ‾=1, the refined formalism does not hold, while Olsen's concave multifractal formalism holds in some cases. This answers a question asked by Olsen (Adv. Math. 116, 1995). We also show that if 0<θ_<θ‾<1, Olsen's concave multifractal formalism does not hold.
Current interest in fractal dimensions of networks is the result of more than a century of previous research on dimensions. Fractal Dimensions of Networks ties the theory and methods for computing fractal dimensions of networks to the classical theory of dimensions of geometric objects.
The goal of the book is to provide a unified treatment of fractal dimensions of sets and networks. Since almost all of the major concepts in fractal dimensions originated in the study of sets, the book achieves this goal by first clearly presenting, with an abundance of examples and illustrations, the theory and algorithms for sets, and then showing how the theory and algorithms have been applied to networks. For example, the book presents the classical theory and algorithms for the box counting dimension for sets, and then presents the box counting dimension for networks. All the major fractal dimensions are studied, e.g., the correlation dimension, the information dimension, the Hausdorff dimension, the multifractal spectrum, as well as many lesser known dimensions. Algorithm descriptions are accompanied by worked examples, with many applications of the methods presented.
· Presentation of a unified view of fractal dimensions and the relationship between computing these dimensions for geometric objects and computing them for networks
· A historical view of the different dimensions, starting with Euclid, presented in a form that is not overly mathematical
· Many applications of the methods are discussed in a broad range of fields: art, biology, cosmology, food processing, marine science, neurology, etc.
· Many examples are provided to illustrate the computational methods
· Includes exercises throughout, ranging in difficulty from simple to research level
In this paper, we introduce a general formalism for the multifractal analysis of one probability measure with respect to another. As examples, we analyze the multifractal structure of quasi-Bernoulli and homogeneous Moran measures.
The aim of this article is to study the behavior of the multifractal packing function \(B_\mu (q)\) under projections in Euclidean space for \(q>1\). We show that \(B_\mu (q)\) is preserved under almost every orthogonal projection. As an application, we study the multifractal analysis of the projections of a measure. In particular, we obtain general results for the multifractal analysis of the orthogonal projections on m-dimensional linear subspaces of a measure \(\mu \) satisfying the multifractal formalism.
In this paper, we establish some density results related to the multifractal generalization of the centered
Hausdorff and packing measures. We also focus on the exact dimensions of locally finite and Borel regular measures. We, then, apply these theories to a class of Moran sets satisfying the strong separation
condition.
The aim of this work is to provide a relationship between the relative multifractal spectra of orthogonal projections of a measure $\mu$ in Euclidean space and those of $\mu$. As an application we study the relative multifractal analysis of the projections of a measure.
The present redaction is mainly an account of a joint work [6] with G. Brown, from the University of New South Wales, and G. Michon, from Dijon University. The multifractal formalism is described, and a setting in which it holds is given, as well as the Michon construction of Gibbs measures.
We are going to show the link between the ϵ (∞) Cantorian space and the Hilbert spaces H (∞) . In particular, El Naschie’s ϵ (∞) is a physical spacetime, i.e. an infinite dimensional fractal space, where time is spacialized and the transfinite nature manifests itself. El Naschie’s Cantorian spacetime is an arena where the physics laws appear at each scale in a self-similar way linked to the resolution of the act of observation. By contrast the Hilbert space H (∞) is a mathematical support, which describes the interaction between the observer and the dynamical system under measurement.The present formulation, which is based on the non-classical Cantorian geometry and topology of spacetime, automatically solves the paradoxical outcome of the two-slit experiment and the so-called particle-wave duality. In particular, measurement (i.e. the observation) is equivalent to a projection of ϵ (∞) in the Hilbert space built on 3+1 Euclidean spacetime. Consequently, the wave–particle duality becomes a mere natural consequence of conducting an experiment in a spacetime with non-classical topological and geometrical structures, while observing and taking measurements in a classical smooth 3+1 Euclidean spacetime. In other words, the experimental fact that a wave-particle duality exists is an indirect confirmation of the existence of ϵ (∞) and a property of the quantum–classical interface. Another direct consequence of the fact that real spacetime is infinite dimensional hierarchical ϵ (∞) is the existence of scaling law R(N), introduced by the author, which generalizes the Compton wavelength. It gives an answer to the problem of segregation of matter at different scales, and shows the role of fundamental constants such as the speed of light and Planck’s constant h in the fundamental lengths scale without invoking the principles of quantum mechanics.
M. Dai and D. Liu [Chaos Solitons Fractals 38, No. 4, 1025–1030 (2008; Zbl 1152.28313)] obtained the formula of local dimensions of some Moran measures on Moran sets in ℝ d under the strong separation condition. In this paper, we prove that the result is still true under the open set condition. Due to the lack of the strong separation condition, our approach is essentially different to that used by Dai and Liu. We also obtain the formulas of the Hausdorff and packing dimensions of the Moran measures and discuss some interesting examples.
In this paper, we shall study the multifractal decomposition behavior for a family of sets E known as Moran fractals. For each value of the parameter α∈(αmin,αmax), we define “multifractal components”Eα of E, and show that they are non-regularity fractals (in the sense of Taylor). By obtaining the new sufficient conditions for the valid multifractal formalisms of non-regularity Moran measures, we give explicit formula for the Hausdorff dimension and Packing dimension of Eα respectively. In particular, we describe a large class of non-regularity Moran measure satisfying the explicit formula.
In this paper we attain the multifractal dimension functions of Moran measure associated with homogeneous Moran fractals and give a sufficient condition which ensures the multifractal spectrum of Moran measure is equal to the Legendre transform of the multifractal dimension functions. As an application of this method, we give an example and a counterexample about the validity of the multifractal formalism of Moran measure.
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.
In this paper we introduce Mohamed El Naschie’s ϵ(∞) Cantorian space–time in connection with stochastic self-similar processes to give a possible explanation of the segregation of the Universe at fixed scale, then we summarize some relevant consequences.
Let X be a metric space and μ a Borel probability measure on X. For q, t ∈ R and E ⊆ X write [formula]Then Hq, tμ and Pq, tμ are Borel measures − Hq, tμ is a multifractal generalization of the centered Hausdorff measure and Pq, tμ is a multifractal generalization of the packing measure. The measures Hq, tμ and Pq, tμ define, for a fixed q, in the usual way a generalized Hausdorff dimension dimqμ(E) and a generalized packing dimension Dimqμ(E) of subsets E of X. We study the functions bμ: q → dimqμ(supp μ), Bμq → Dimqμ(supp μ) and their relation to the so-called multifractal spectra functions of μ: [formula] We prove, among other things, that fμ(Fμ) is bounded from above by the Legendre transform of bμBμ) and that equality holds for graph directed self-similar measures and "cookie-cutter" measures. Finally we discuss the connection with generalized Rényi dimensions.
We propose a description of normalized distributions (measures) lying upon possibly fractal sets; for example those arising in dynamical systems theory. We focus upon the scaling properties of such measures, by considering their singularities, which are characterized by two indices: α, which determines the strength of their singularities; and f, which describes how densely they are distributed. The spectrum of singularities is described by giving the possible range of α values and the function f(α). We apply this formalism to the 2∞ cycle of period doubling, to the devil’s staircase of mode locking, and to trajectories on 2-tori with golden-mean winding numbers. In all cases the new formalism allows an introduction of smooth functions to characterize the measures. We believe that this formalism is readily applicable to experiments and should result in new tests of global universality.
This book covers analysis on fractals, a developing area of mathematics which focuses on the dynamical aspects of fractals, such as heat diffusion on fractals and the vibration of a material with fractal structure. The book provides a self-contained introduction to the subject, starting from the basic geometry of self-similar sets and going on to discuss recent results, including the properties of eigenvalues and eigenfunctions of the Laplacians, and the asymptotical behaviors of heat kernels on self-similar sets. Requiring only a basic knowledge of advanced analysis, general topology and measure theory, this book will be of value to graduate students and researchers in analysis and probability theory. It will also be useful as a supplementary text for graduate courses covering fractals.
In this paper we establish the complete multifractal formalism for equilibrium measures for Hlder continuous conformal expanding maps andexpanding Markov Moran-like geometric constructions. Examples include Markov maps of an interval, beta transformations of an interval, rational maps with hyperbolic Julia sets, and conformal toral endomorphisms. We also construct a Hlder continuous homeomorphism of a compact metric space with an ergodic invariant measure of positive entropy for which the dimension spectrum is not convex, and hence the multifractal formalism fails.
"...a blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) ...and the illustrations include many superb examples of computer graphics that are works of art in their own right." Nature
This paper discusses possible connections between quite different and partly rather remote disciplines such as hyper-dimensional sphere-packing lattices, superstring theory and the Cantorianfractal spacetime approach to particle physics. Furthermore, relations to fractional diffusion, Nagasawa's diffusion, quantum chaos and fractal gravity are briefly considered.
We introduce a general formalism for the multifractal analysis of one probability measure with respect to another. As an example, we analyse the multifractal structure of one graph directed self-conformal measure with respect to another.
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