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Singularity Spectrum of Generic α-Hölder Regular Functions After Time Subordination

  • Zoltán BuczolichEmail author
  • Stéphane Seuret
Article

Abstract

A question of Yves Meyer motivated the research concerning “time” subordinations of real functions. Denote by \({\mathcal {B}}_{1}^{\alpha}\) the metric space of functions with Lipschitz constant 1 defined on [0,1], equipped with the complete metric defined via the supremum norm. Given a function \(g\in {\mathcal {B}}_{1}^{\alpha}\) one obtains a time subordination of g simply by considering the composite function Z=gf, where f∈ℳ:={f:f(0)=0, f(1)=1 and f is a continuous nondecreasing function on [0,1]}. The metric space \(\mathcal {E}^{\alpha}=\mathcal {M}\times {\mathcal {B}}_{1}^{\alpha}\) equipped with the product supremum metric is a complete metric space. In this paper for all α∈[0,1) multifractal properties of gf are investigated for a generic (typical) element (f,g)∈ℰ α . In particular we determine the generic Hölder singularity spectrum of gf.

Keywords

Hölder regularity Time subordination Hausdorff measure Hausdorff dimension 

Mathematics Subject Classification (2000)

28A80 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Eötvös Loránd UniversityBudapestHungary
  2. 2.LAMA, CNRS UMR 8050Université Paris-EstCréteil CedexFrance

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