Multifractal Hausdorff Research Articles

SUBSETS OF POSITIVE AND FINITE MULTIFRACTAL MEASURES

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.

On the multifractal measures and dimensions of image measures on a class of Moran sets

On a Moran set meeting the strong separation requirement, we examine a family of multifractal 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.

On the vectorial multifractal analysis in a metric space

<abstract><p>Multifractal analysis is typically used to describe objects possessing some type of scale invariance. During the last few decades, multifractal analysis has shown results of outstanding significance in theory and applications. In particular, it is widely used to characterize the geometry of the singularity of a measure $ \mu $ or to study the time series, which has become an important tool for the study of several natural phenomena. In this paper, we investigate a more general level set studied in multifractal analysis. We use functions defined on balls in a metric space and that are Banach valued which is more general than measures used in the classical multifractal analysis. This is done by investigating Peyrière's multifractal Hausdorff and packing measures to study a relative vectorial multifractal formalism. This leads to results on the simultaneous behavior of possibly many branching random walks or many local Hölder exponents. As an application, we study the relative multifractal binomial measure in symbolic space $ \partial {\mathcal A} $.</p></abstract>

On the equivalence of multifractal measures on Moran sets

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.

Multifractal dimensions for projections of measures

In this paper, we calculate the multifractal Hausdorff and packing dimensions of Borel probability measures and study their behaviors under orthogonal projections. In particular, we try through these results to improve the main result of M. Dai in \cite{D} about the multifractal analysis of a measure of multifractal exact dimension.

On the multifractal analysis of measures in a probability space

In this paper, we calculate the relative multifractal Hausdorff and packing dimensions of measures in a probability space. Also, we obtain the analogue of Frostman’s lemma in a probability space for a relative multifractal Hausdorff measure. In the same way, there is a valid result for the relative multifractal packing pre-measure. Furthermore, we obtain the representations of the functions b and B by means of the analogue of Frostman’s lemma, and we provide a technique for showing that E is a (q,μ)-fractal with respect to ν. In addition, we suggest new proofs of theorems on the relative multifractal formalism in a probability space. They yield results even at a point q for which the multifractal functions b(q) and B(q) differ.

The relative multifractal densities: A review and application

We are concerned with some density estimations for relative multifractal measures. We tried, through this article, to improve many known density results. The results in this paper establish the connections with various relative multifractal measures and generalize many known results about density theorems. As an application, we mainly discuss the equivalence of the relative multifractal Hausdorff and packing measures on cookie-cutterlike sets satisfying the strong separation condition.

Remarks on the mutual singularity of multifractal measures

In the present work, we study the mutual singularity of multifractal Hausdorff and packing measures which provide a positive answer to Olsen’s questions in a more general framework. Our main results apply to a family of measures supported by the full 5-adic grid of [0, 1], namely the quasi-Bernoulli measures.

Another example of the mutual singularity of multifractal measures

We propose an example for which the multifractal Hausdorff and packing measures are mutually singular.

Projections of measures with small supports

Abstract In this paper, we use a characterization of the mutual multifractal Hausdorff dimension in terms of auxiliary measures to investigate the projections of measures with small supports.

The relative multifractal analysis, review and examples

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 Holder 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.

Some New Characterizations of Olsen’s Multifractal Functions

The aim of this work is to provide a representation of the multifractal Hausdorff and packing dimensions of compact sets in the Euclidean space in terms of the lower and upper local $$(q,\mu )$$ -dimensions of a probability measure on $$\mathbb {R}^n$$ , respectively. In addition, a relationship is established allowing to determine the Olsen’s multifractal functions by means of the corresponding measures’ versions. The multifractal functions are defined as the supremum over the lower and upper multifractal dimensions of all Borel probability measures.

On the mutual singularity of multifractal measures

<p style='text-indent:20px;'>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.</p>

Relative multifractal box-dimensions

Given two probability measures ? and ? on Rn. We define the upper and lower relative multifractal box-dimensions of the measure ? with respect to the measure ? 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.

On the strong regularity with the multifractal measures in a probability space

We prove a decomposition theorem of Besicovitch’s type for the relative multifractal Hausdorff measure and packing measure in a probability space. By obtaining a new necessary condition for the strong regularity with the multifractal measures in a more general framework, we extend in this paper the density theorem of Dai and Li (A multifractal formalism in a probability space. Chaos Solitons Fractals 27:57–73, 2006). In particular, this result is more refined than those found in Dai and Taylor (Defining fractal in a probability space. Ill J Math 38:480–500, 1994).

Some results about the regularities of multifractal measures

In this paper, we generelize the Olsen's density theorem to any measurable set, allowing us to extend the main results of H.K. Baek in \big(Proc. Indian Acad. Sci. (Math. Sci.) Vol. {\bf118}, (2008), pp. 273-279.\big). In particular, we tried through these results to improve the decomposition theorem of Besicovitch's type for the regularities of multifractal Hausdorff measure and packing measure.

Empirical multifractal moment measures of self-similar measure for q < 0*

For q ≥ 0, Olsen [1] has attained the exact rate of convergence of the L q -spectrum of a self-similar measure and showed that the so-called empirical multifractal moment measures converges weakly to the normalized multifractal measures. Unfortunately, nothing is known for q < 0. Indeed, the problem of analysing the L q - spectrum for q < 0 is generally considered significantly more difficult since the L q -spectrum is extremely sensitive to small variations of μ for q < 0. In [2] we showed that self-similar measures satisfying the Open Set Condition (OSC) are Ahlfors regular and, using this fact, we obtained the exact rate of convergence of the L q -spectrum of a self-similar measure satisfying the OSC for q < 0. In this paper, we apply the results from [2] to show the empirical multifractal q’th moment measures of self-similar measures satisfying the OSC converges weakly to the normalized multifractal Hausdorff measures for q < 0.

Regularities of multifractal measures

First, we prove the decomposition theorem for the regularities of multifractal Hausdorff measure and packing measure in ”d. This decomposition theorem enables us to split a set into regular and irregular parts, so that we can analyze each separately, and recombine them without affecting density properties. Next, we give some properties related to multifractal Hausdorff and packing densities. Finally, we extend the density theorem in [6] to any measurable set.

Multifractal dimension inequalities in a probability space

Let ν be a probability measure on Ω . We define the upper and lower multifractal box dimension (the measure ν with respect to μ ) on a probability space and investigate the relation between the multifractal box dimension and the multifractal Hausdorff dimension, the multifractal pre-packing dimension . Then, we generalize the dimension inequalities of multifractal Hausdorff measures and multifractal packing measures in a probability space.

Hausdorff measures, dimensions and mutual singularity

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.