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In this paper, we attain the multifractal Hewitt–Stromberg dimension functions of Moran measures associated with homogeneous Moran fractals and show that the multifractal Hewitt–Stromberg measures are mutually singular for which the multifractal functions do not necessarily coincide. In particular, we give a positive answer to questions posed in Attia and Selmi (J Geom Anal 31:825-862, 2021) and discuss some interesting examples.

This paper studies the behavior of the lower and upper multifractal Hewitt–Stromberg functions under slices onto (n-m){(n-m)}-dimensional subspaces. More precisely, we discuss the relationship between the multifractal Hewitt–Stromberg functions of a compactly supported Borel probability measure and those of slices or sections of the measure. In addition, we prove that if μ has a finite m-energy and qlies in a certain somewhat restricted interval, then these functions satisfy the expected adding of co-dimensions formula.

AbstractIn this paper, a generalized fractal dimension is introduced based on a combination of the topological dimension, conformal dimension, and the ϕ-conformal dimension. Lower and upper bound estimates of the new variant of dimension are provided for the case of the Sierpinski carpet fractal set. Moreover, the equality of such bounds is proved for a large class of the basic measure ϕ.

AbstractWe 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.

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 fixed point theory is one of the most essential techniques of applicable mathematics for solving many realistic problems to get a unique solution by using the well known Banach contraction principle. It has paved the ways for numerous extensions, generalization and development of the theory of fixed points in very diverse settings. Our intention in the present paper is to study the Kannan contraction maps defined on a controlled metric space. The generalization of the fixed point theorem for Kannan contraction on controlled metric space is investigated in this paper. We construct an iterated function system called Controlled Kannan Iterated Function System (CK-IFS) with Kannan contraction maps in a controlled metric space and use it to develop a new kind of invariant set, which is called a Controlled Kannan Attractor or Controlled Kannan Fractal (CK-Fractal). Subsequently, the collage theorem for controlled Kannan fractal is also proved. The multivalued fractals are also constructed in the controlled metric space using Kannan and Reich-type contraction maps. The newly developing iterated function system and fractal set in the controlled metric space can provide a novel direction in the fractal theory. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notions of the mean packing dimension and mean pseudo-packing dimension. The product formulas for the mean Hausdorff dimension, the mean packing dimension, the mean pseudo-packing dimension and the metric mean dimension are established. [ABSTRACT FROM AUTHOR]

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 regularitywith 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).

• One of the most important single results for the Lebesgue measure is the Lebesgue density theorem. • Marstrand's Theorem asserts that density in the Euclidean space can exist only if the dimension is an integer. • Tricot established related results in a more general context. • In this work, we prove a decomposition theorem of Besicovitch's type for the regularities of general Hausdorff and packing functions. • We try to formulate another version of Tricot's density theorem. We prove a decomposition theorem of Besicovitch's type for the regularities of general Hausdorff and packing functions. As an application, we try to formulate another version of Tricot's density theorem. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The aim of this work is to provide a relationship between the relative multifractal spectra of orthogonal projections of a measure μ in Euclidean space and those of μ . As an application we study the relative multifractal analysis of the projections of a measure. [ABSTRACT FROM AUTHOR]

In this study, we will introduce an innovative and comprehensive multifractal framework, substantiating counterparts to the classical findings in multifractal analysis and We embark on an exploration of the mutual singularity existing between the broader multifractal Hausdorff and packing measures within an expansive framework. An exemplar of this framework involves the application of the " second-order " multifractal formalism to our core results, elucidating a realm of compact subsets within a self-similar fractal structure — a familiar illustration of an infinite-dimensional metric space. Within this context, we provide estimations for both the overall Hausdorff and packing dimensions. It is noteworthy that these outcomes offer novel validations for theorems underpinning the multifractal formalism, rooted in these comprehensive multifractal measures. Furthermore, these findings remain valid even at points q where the multifractal functions governing Hausdorff and packing dimensions diverge. Additionally, we introduce the concepts of lower and upper relative multifractal box dimensions, accompanied by the general Rényi dimensions. A comparison is drawn between these dimensions and the general multifractal Hausdorff dimension, along with the general multifractal pre-packing dimension. Finally, we establish density conclusions pertaining to the multifractal extension of the centered Hausdorff and packing measures. Specifically, we unveil a decomposition theorem akin to Besicovitch's theorem for these measures. This theorem facilitates a division into two components – one characterized as regular and the other as irregular – thus enabling a targeted analysis of each segment. Following this dissection, we seamlessly reintegrate these segments while preserving their intrinsic density attributes. The impetus behind investigating these comprehensive measures arises when a set E holds either a conventional Hausdorff measure of zero or infinity; In such cases, we can identify a function ϕ s that bestows a positive and finite general Hausdorff measure upon the set E , and studies the dimensions of infinitely dimensional sets. • We are presenting an innovative multifractal framework that builds upon the conventional findings in multifractal analysis. • We demonstrate an example by applying the "second-order" multifractal concept to our core results. • We provide estimations for both the overall Hausdorff dimension and packing dimension. • We introduce the lower and upper relative multifractal box dimensions, along with the general Rényi dimensions. [ABSTRACT FROM AUTHOR]

• In our present work, we are looking for a slight modification of the generalized multifractal analysis. Instead of studying sets of points with a local dimension given with respect to Lebesgue measure, we study sets of points with a local dimension given with respect to an non-atomic probability measure ν and checks an auxialary condition. • We have found that the class of measures checking specific condition, which will be specified later, and responding to the multifractal formalism thus considered is large, which we can cite: quasi-Bernoulli measures, one-dimensional measures and homogeneous Moran measures. Actually, it is very natural to study this formalism of multifractal analysis for what is differs slightly from which introduced by Cole. The difference between the two types is that we used centered ν − δ - coverings rather than centered − δ -coverings. • We should not that more motivations and examples related to these concepts, will be discussed. 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. [ABSTRACT FROM AUTHOR]