Search

Let ν be a Borel probability measure on ℝ d {\mathbb{R}^{d}} and q , t ∈ ℝ {q,t\in\mathbb{R}} . This study takes a broad approach to the multifractal and fractal analysis problem and proposes an intrinsic definition of the general Hausdorff and packing measures by taking into account sums of the type ∑ i h - 1 ⁢ ( q ⁢ h ⁢ ( ν ⁢ ( B ⁢ ( x i , r i ) ) ) + t ⁢ g ⁢ ( r i ) ) \sum_{i}h^{-1}(qh(\nu(B(x_{i},r_{i})))+tg(r_{i})) for some prescribed functions h and g. The aim of this paper is to study the descriptive set-theoretic complexity and measurability of these measures and related dimension maps. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The properties of various fractal and multifractal measures and dimensions have been under extensive study in the real-line and higher-dimensional Euclidean spaces. In non-Euclidean spaces, it is often impossible to construct non-trivial self-similar or self-conformal sets, etc. We consider in the present paper the proper way to phrase the definitions for use in general metric spaces. We investigate the relative Hausdorff measures and the relative packing measures defined in a separable metric space. We give some product inequalities which are a consequence of a new version of density theorems for these measures. Moreover, we prove that and can be expressed as Henstock-Thomson variation measures. The question of the weak-Vitali property arises in this context. [ABSTRACT FROM AUTHOR]

In this work, we focus on the centered Hausdorff measure, the packing measure, and the Hewitt–Stromberg measure that determines the modified lower box dimension Moran fractal sets. The equivalence of these measures for a class of Moran is shown by having a strong separation condition. We give a sufficient condition for the equality of the Hewitt–Stromberg dimension, Hausdorff dimension, and packing dimensions. As an application, we obtain some relevant conclusions about the Hewitt–Stromberg measures and dimensions of the image measure of a τ -invariant ergodic Borel probability measures. Moreover, we give some statistical interpretation to dimensions and corresponding geometrical measures. [ABSTRACT FROM AUTHOR]

We are interested in the behavior of relative multifractal box-dimensions, density results and multifractal dimensions under projections onto a lower-dimensional linear subspace. The results in this paper establish the connections with various multifractal and fractal dimensions and their projections, and generalize many known results about Hausdorff and packing measures and dimensions of a subset of ℝn. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to investigate the average general fractal dimensions of typical compact metric spaces within the Gromov-Hausdorff space, using the Gromov-Hausdorff metric. As an application of the main results, we demonstrate that a typical compact metric space Σ exhibits such irregularity that the lower and upper average general Hewitt-Stromberg dimensions, as well as the general box dimensions corresponding to all higher-order Hölder and Cesàro averages, diverge significantly. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to present a comprehensive exploration of general multifractal dimensions of Borel probability measures, with a specific emphasis on the diverse methodologies available for their estimation or computation. We prove that these general dimensions enable us to compare a probability measure ϑ with the general multifractal Hausdorff or packing measure. Also, we aim to measure the degree of regularity or singularity of ϑ , and we examine the correlations between various approaches to defining the general multifractal exact dimension of Borel probability measures, using fractal analysis of the essential supports of these measures as the foundation. [ABSTRACT FROM AUTHOR]

The aim of this article is to study the behavior of the relative multifractal spectrum under projections. First of all, we depict a relationship between the mutual multifractal spectra of a couple of measures (μ, ν) and its orthogonal projections in Euclidean space. As an application, we improve Svetova's results in [46] and study the mutual multifractal analysis of the projections of measures. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 v-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. [ABSTRACT FROM AUTHOR]

In the present work, we are concerned with some multifractal dimensions estimations of vector-valued measures in the framework of the so-called mixed multifractal analysis. We precisely consider some Borel probability measures that are no longer Gibbs and introduce some mixed multifractal generalizations of Hausdorff and packing dimensions of measures in a framework of relative mixed multifractal analysis. As an application, we are interested in the '-unidimensionality of those measures and to the calculus of its mixed multifractal Hausdorff and packing dimensions. In particular, we give a necessary and sufficient condition for the existence of the '-mixed multifractal Hausdorff and packing dimensions of a Borel probability measure. Finally, concrete examples satisfying the above property are developed. [ABSTRACT FROM AUTHOR]

Conditional aggregation operators are defined by the Choquet integral and the Sugeno integral with respect to a monotone set function that assesses positive measure of the conditioning set. General Hausdorff and packing measures are introduced and examples of infinite s-sets with positive and finite generalized Hausdorff and packing measures are constructed and their fractal dimensions are compared. Coherent upper conditional previsions on the linear space of all Choquet integrable random variables are defined by the Choquet integral with respect to the general Hausdorff and packing measures when the conditioning event has positive and finite generalized Hausdorff and packing measures in its respective fractal dimensions. Conditional aggregation operators are defined by the Sugeno integral with respect to general Hausdorff and packing measures on the class of all Sugeno integrable random variables. Actually, the general Hausdorff and packing dimensions are proven to be the Sugeno integral with respect to the Lebesgue measure of the general Hausdorff and packing measures respectively. [ABSTRACT FROM AUTHOR]

We propose an example for which the multifractal Hausdorff and packing measures are mutually singular. [ABSTRACT FROM AUTHOR]

By introducing a novel expression for the Hausdorff and packing measures with respect to gauge functions, we establish a connection between different approaches to defining the exact Hausdorff and packing dimensions of measures. Furthermore, we define the upper and lower Hausdorff and packing dimensions of a Borel probability measure and demonstrate that these dimensions can be represented using an entropy formula. This paper also addresses an intriguing question raised multiple times regarding the structural representation of singular probability measures. We provide an affirmative response to this question by utilizing characteristic measures. [ABSTRACT FROM AUTHOR]