The Multifractal Spectra of V-Statistics

  • Ai-hua Fan
  • Jörg Schmeling
  • Meng Wu
Part of the Trends in Mathematics book series (TM)


Let (X, T) be a topological dynamical system and let Φ : X r  → ℝ be a continuous function on the product space X r  = X ×⋯ ×X (r ≥ 1). We are interested in the limit of V-statistics taking Φ as kernel:
$$\lim\limits_{n\rightarrow \infty }{n}^{-r}\displaystyle\sum\limits_{ 1\leq i_{1},\cdots \,,i_{r}\leq n}\Phi ({T}^{i_{1} }x,\cdots \,,{T}^{i_{r} }x).$$
The multifractal spectrum of topological entropy of the above limit is expressed by a variational principle when the system satisfies the specification property. Unlike the classical case (r = 1) where the spectrum is an analytic function when Φ is Hölder continuous, the spectrum of the limit of higher-order V-statistics (r ≥ 2) may be discontinuous even for very nice kernel Φ.


Hausdorff Dimension Probability Vector Topological Entropy Multifractal Analysis Multifractal Spectrum 
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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.LAMFA, UMR 7352 CNRSUniversity of Picardie JulesAmiens CedexFrance
  2. 2.Mathematics Centre for Mathematical Sciences, Lund Institute of TechnologyLund UniversityLundSweden
  3. 3.Laboratoire Amiénois. de Mathématique Fondamentale et Appliquée, UMR 7352 CNRSUniversity of PicardieAmiens CedexFrance

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