Multifractal Hausdorff Research Articles

A multifractal formalism in a probability space

Abstract In this paper, we mainly research a multifractal formalism in a probability space.The first objective of this paper is to define a multifractal generalisation of the Hausdorff measure and the packing measure. We explore the properties of the multifractal Hausdorff measure and the multifractal packing measure in a probability space, and invesigate the relative results of multifractal spectra functions in a probability space. We also describe a sufficient condion and a necessary one of validity for the multifractal formalism in a probability space.

Multifractal Variation Measures and Multifractal Density Theorems

In this paper we show that the multifractal Hausdorff measure and multifractal packing measure introduced by Olsen and Peyriere can be expressed as Henstock-Thomson \lq\lq variation measures. As an application we prove a density theorem for these two measures that extends results by Edgar and is more refined than those found in [Ol1].

THE MULTIFRACTAL HAUSDORFF AND PACKING MEASURE OF GENERAL SIERPINSKI CARPETS

In this paper, authors study the properties of multifractal Hausdorff and packing measures for a class of self-affine sets and use them to study the multifractal properties of general Sierpinski carpet E, and they get that the multifractal Hausdorff and packing measure are mutual singular, when they are restricted on some subsets of E.

Dimension inequalities of multifractal Hausdorff measures and multifractal packing measures

Let be a Borel probability measure on R . We study the Hausdorff dimension and the packing dimension of the multifractal Hausdorff measurehq;t and the multifractal packing measure p q;t introduced in [L. Olsen, A multifractal formalism, Advances in Mathematics 116 (1995), 82^196]. Let b denote the multifractal Hausdorff dimension function and let B denote the multifractal packing dimension function introduced in [Olsen, op_ cit_]. For a fixed q 2 R, we obtain bounds for the Hausdorff dimension and the packing dimension of hq;b q† and p q;B q† in terms of the subdifferential of b and B at q. For q ˆ 1, our result reduces to yD‡B 1† lim inf r&0 log B x; r† log r lim sup r&0 log B x; r† log r yDyB 1† for -a.a. x † where DyB 1† and D‡B 1† denote the left and right derivative of B at 1. Inequality ( ) improves a similar result obtained independently by Y: Heurteaux and S:-Z: Ngai. It follows from ( ) that if the mulifractal box dimension spectrum (or Lq spectrum) of is differentiable at 1 then y 0 1† equals the entropy dimension (or information dimension) of . This result has been conjectured in the physics literature and proved rigorously in certain special cases.

The Average (n) Sphere Spans a Four-Dimensional Manifold

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.

Multifractal dimensions of product measures

AbstractWe study the multifractal structure of product measures. for a Borel probability measure μ and q, t Є , let and denote the multifractal Hausdorff measure and the multifractal packing measure introduced in [O11] Let μ be a Borel probability merasure on k and let v be a Borel probability measure on t. Fix q, s, t Є . We prove that there exists a number c > 0 such that for E ⊆k, F ⊆l and Hk+l provided that μ and ν satisfy the so-called Federer condition.Using these inequalities we give upper and lower bounds for the multifractal spectrum of μ × ν in terms of the multifractal spectra of μ and ν