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Multifractal analysis for projections of Gibbs and related measures

Published online by Cambridge University Press:  26 May 2010

JULIEN BARRAL  and
IMEN BHOURI
JULIEN BARRAL
Affiliation:
LAGA (UMR 7539), Institut Galilée, 99 av. Jean-Baptiste Clément, 93430 Villetaneuse, France (email: barral@math.univ-paris13.fr)
IMEN BHOURI
Affiliation:
Département de Mathématiques, Faculté des Sciences de Monastir, 5019 Monastir, Tunisia (email: imen.bhouri@fsm.rnu.tn)
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Abstract

Let n>m≥1 be two integers. At first we obtain general results for the multifractal analysis of the orthogonal projections on m-dimensional linear subspaces of singular measures μ on ℝn satisfying the multifractal formalism. The results hold for γn,m-almost every such subspace, where γn,m is the uniform measure on the Grassmannian manifold Gn,m. Let μ be such a measure and suppose that its upper Hausdorff dimension is less than or equal to m. Let I stand for the interval over which the singularity spectrum of μ is increasing. We prove that there exists a non-trivial subinterval of I such that for every , for γn,m-almost every m-dimensional subspace V, the multifractal formalism holds at α for μV, the orthogonal projection of μ on V. Moreover, in some cases the result is optimal in the sense that the interval is maximal in I. Also, we determine the Lq-spectrum τμV(q) on the minimal interval J necessary to recover the singularity spectrum of μV over as the Legendre transform of τμV. The interval J and the function τμV(q) do not depend on V, and τμV(q) can differ from τμ on a non-trivial interval. For Gibbs measures and some of their discrete counterparts, we show the stronger uniform result: for γn,m-almost every m-dimensional subspace V, the multifractal formalism holds for μV over the whole interval . As an application, we obtain a part of the singularity spectrum of some self-similar measures on attractors of iterated function systems which do not satisfy the weak separation condition.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 3 , June 2011 , pp. 673 - 701
Copyright
Copyright © Cambridge University Press 2010

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