The Validity of the Multifractal Formalism: Results and Examples

Nasr, Fathi Ben; Bhouri, Imen; Heurteaux, Yanick

doi:10.1006/aima.2001.2025

Cited by 70 publications

(60 citation statements)

References 15 publications

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“…Of course, this way is not an option in many applications where the structure of the measure is not known in advance. For more information on the L q -spectrum and the singularity spectrum we refer the reader to [1,2,5,10,11,19,28,31,32,37,38,39,41].…”

Section: Introductionmentioning

confidence: 99%

Phase transitions for the multifractal analysis of self-similar measures

Testud

2006

Nonlinearity

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Abstract. We are interested in the multifractal analysis of a class of self-similar measures with overlaps. This class, for which we obtain explicit formulae for the L qspectrum τ (q) as well as the singularity spectrum f (α), is sufficiently large to point out new phenomena in the multifractal structure of self-similar measures. We show that, unlike the classical quasi-Bernoulli case, the L q -spectrum τ (q) of the measures studied can have an arbitrarily large number of non-differentiability points (phase transitions). These singularities occur only for the negative values of q and yield to measures that do not satisfy the usual multifractal formalism. The weak quasi-Bernoulli property is the key point of most of the arguments.

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Section: Introductionmentioning

confidence: 99%

Phase transitions for the multifractal analysis of self-similar measures

Testud

2006

Nonlinearity

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“…For the framework of multifractal analysis of general measures, one may see, e.g., [4,7,8,21,24,25,26,37,39,42,44]. It is well known that if µ is the self-similar measure defined by a family of contractive similitudes {S j } j=1 which satisfies the open set condition [28], τ (q) can be calculated by an explicit analytic formula and µ satisfies the complete multifractal formalism (see [8,37]).…”

Section: Introductionmentioning

confidence: 99%

Lyapunov exponents for products of matrices and multifractal analysis. Part I: Positive matrices

Ding

2003

Isr. J. Math.

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Abstract. We continue the study in [11,14] on the upper Lyapunov exponents for products of matrices. Here we consider general matrices. In general, the variational formula about Lyapunov exponents we obtained in part I does not hold in this setting. Anyway we focus our interest on a special case where the matrix function M (x) takes finite values M1, . . ., Mm. In this case we prove the variational formula under an additional irreducibility condition. This extends a previous result of the author and Lau [14]. As an application, we prove a new multifractal formalism for a certain class of self-similar measures on R with overlaps. More precisely, let µ be the self-similar measure on R generated by a family of contractive similitudes {Sj = ρx + bj} j=1 which satisfies the finite type condition. Then we can construct a family (finite or countably infinite) of closed intervals {Ij}j∈Λ with disjoint interiors, such that µ is supported on j∈Λ Ij and the restricted measure µ|I j of µ on each interval Ij satisfies the complete multifractal formalism. Moreover the dimension spectrum dimH E µ| I j (α) is independent of j.

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“…For the framework of multifractal analysis of general measures, one may see, e.g., [4,7,8,11,22,24,25,37,39,41,44]. It is well-known that if µ is the self-similar measure defined by a family of contractive similitudes {S j } j=1 which satisfies the open set condition [27], τ (q) can be calculated by an explicit analytic formula and µ satisfies the complete multifractal formalism (see [8,37]).…”

Section: Introductionmentioning

confidence: 99%

Lyapunov exponents for products of matrices and multifractal analysis. Part II: General matrices

Ding

2009

Isr. J. Math.

157

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We continue the study in [15,18] on the upper Lyapunov exponents for products of matrices. Here we consider general matrices. In general, the variational formula about Lyapunov exponents we obtained in part I does not hold in this setting. In any case, we focus our interest on a special case where the matrix function M (x) takes finite values M 1 , . . ., Mm. In this case, we prove the variational formula under an additional irreducibility condition. This extends a previous result of the author and Lau [18]. As an application, we prove a new multifractal formalism for a certain class of self-similar measures on R with overlaps. More precisely, let µ be the self-similar measure on R generated by a family of contractive similitudes {S j = ρx + b j } j=1 which satisfies the finite type condition. Then we can construct a family (finite or countably infinite) of closed intervals {I j } j∈Λ with disjoint interiors, such that µ is supported on j∈Λ I j and the restricted measure µ| I j of µ on each interval I j satisfies the complete multifractal formalism. Moreover, the dimension spectrum dim H E µ| I j (α)is independent of j.

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The Validity of the Multifractal Formalism: Results and Examples

Cited by 70 publications

References 15 publications

Phase transitions for the multifractal analysis of self-similar measures

Phase transitions for the multifractal analysis of self-similar measures

Lyapunov exponents for products of matrices and multifractal analysis. Part I: Positive matrices

Lyapunov exponents for products of matrices and multifractal analysis. Part II: General matrices

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