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

In the present work, we give a new multifractal formalism for which the classical multifractal formalism does not hold. We precisely introduce and study a multifractal formalism based on the Hewitt–Stromberg measures and that this formalism is completely parallel to Olsen’s multifractal formalism which is based on the Hausdorff and packing measures.

A Multifractal Formalism for Hewitt–Stromberg Measures

In this paper, we study 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.

Multifractal dimensions for projections of measures

We are interested in the behavior of Billingsley dimensions under projections onto a lower dimensional linear subspace. The results in this paper establish the connections with various dimensions o...

On the effect of projections on the Billingsley dimensions

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

We show, under an iterative condition, generalizing that of Ambrosetti–Rabinowitz and using a variational method, the existence of a T-periodic solution for the autonomous superquadratic second order Hamiltonian system with even potential 

$$\begin{aligned} \ddot{z} +V'(z)=0, \qquad z\in \mathbb R^N, \qquad N\in \mathbb N^* \end{aligned}$$

for any prescribed period $T>0$. Moreover, under a certain symmetry condition, such a solution possesses T or T / 3 as its minimal period.

Chouhaïd Souissi

Papers

Some density results of relative multifractal analysis

Generalized Ambrosetti–Rabinowitz condition for minimal period solutions of autonomous Hamiltonian systems

RETRACTED ARTICLE: Minimal Period Symmetric Solutions for Some Hamiltonian Systems Via the Nehari Manifold Method