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The multifractal nature of heterogeneous sums of Dirac masses

Published online by Cambridge University Press:  01 May 2008

JULIEN BARRAL  and
STÉPHANE SEURET
JULIEN BARRAL
Affiliation:
INRIA Rocquencourt, B.P. 105, 78153 Le Chesnay Cedex, France. e-mail: julien.barral@inria.fr
STÉPHANE SEURET
Affiliation:
INRIA Rocquencourt, B.P. 105, 78153 Le Chesnay Cedex, France. e-mail: julien.barral@inria.fr
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Abstract

This paper investigates the natural problem of performing the multifractal analysis of heterogeneous sums of Dirac masses where (xn)n≥0 is a sequence of points in [0, 1]d and (wn)n≥0 is a positive sequence of weights such that Σn≥0wn < ∞. We consider the case where the points xn are roughly uniformly distributed in [0, 1]d, and the weights wn depend on a random self-similar measure μ, a parameter ρ ∈ (0, 1], and a sequence of positive radii (λn)n≥1 converging to 0 in the following way The measure ν has a rich multiscale structure. The computation of its multifractal spectrum is related to heterogeneous ubiquity properties of the system {(xn,λn)n with respect to μ.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 144 , Issue 3 , May 2008 , pp. 707 - 727
Copyright
Copyright © Cambridge Philosophical Society 2008

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