Geometry of sets and measures in euclidean spaces fractals and rectifiability

Convergence of Probability Measures. By Patrick Billingsley. Pp. xii, 253. 117s. 1968. (John Wiley & Sons, Inc.)

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.

Some density results of relative multifractal analysis

Regularities of multifractal hewitt-stromberg measures

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).

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

The aim of this article is to study the behaviour 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.

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On the projections of the multifractal packing dimension for q>1

https://comptes-rendus.academie-sciences.fr/mathematique/item/10.5802/crmath.21.pdf

On the Billingsley dimension of Birkhoff average in the countable symbolic space

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

On the mutual singularity of Hewitt–Stromberg measures for which the multifractal functions do not necessarily coincide

https://comptes-rendus.academie-sciences.fr/mathematique/article/CRMATH_2020__358_8_939_0.pdf

Appendix to the paper “On the Billingsley dimension of Birkhoff average in the countable symbolic space”

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 φ-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 φ-mixed multifractal Hausdorff and packing dimensions of a Borel probability measure. Finally, concrete examples satisfying the above property are developed.

/pdf/multifractal-dimensions-of-vector-valued-non-gibbs-measures-3ytymvswnn.pdf

Bilel Selmi

Papers

On the projections of the multifractal packing dimension for q>1

On the Billingsley dimension of Birkhoff average in the countable symbolic space

On the mutual singularity of Hewitt–Stromberg measures for which the multifractal functions do not necessarily coincide

Appendix to the paper “On the Billingsley dimension of Birkhoff average in the countable symbolic space”

Multifractal dimensions of vector-valued non-Gibbs measures