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On the Hausdorff Dimension of Graphs of Prevalent Continuous Functions on Compact Sets

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Further Developments in Fractals and Related Fields

Part of the book series: Trends in Mathematics ((TM))

Abstract

Let K be a compact set in ℝ d with positive Hausdorff dimension. Using a fractional Brownian motion, we prove that in a prevalent set of continuous functions on K, the Hausdorff dimension of the graph is equal to \(\dim _{\mathcal{H}}(K) + 1\). This is the largest possible value. This result generalizes a previous work due to J.M. Fraser and J.T. Hyde which was exposed in the conference Fractals and Related Fields II. The case of α-Hölderian functions is also discussed.

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Authors and Affiliations

  1. Laboratoire de Mathématiques, Clermont Université, Université Blaise Pascal, 10448, F-63000, Clermont-Ferrand, France

    Frédéric Bayart & Yanick Heurteaux

  2. Laboratoire de Mathématiques, CNRS, Complexe Scientifique des Cézeaux, UMR 6620, F-63177, Aubiere Cedex, France

    Frédéric Bayart & Yanick Heurteaux

Authors
  1. Frédéric Bayart
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  2. Yanick Heurteaux
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Corresponding author

Correspondence to Frédéric Bayart .

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Editors and Affiliations

  1. , LAGA, Université Paris 13, avenue J.B. Clément 99, Villetaneuse, 93430, France

    Julien Barral

  2. , Laboratoire d'Analyse et de Mathématique, Université Paris-Est Créteil - Val de Ma, Avenue du Général de Gaulle 61, Créteil Cedex, 94010, France

    Stéphane Seuret

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Bayart, F., Heurteaux, Y. (2013). On the Hausdorff Dimension of Graphs of Prevalent Continuous Functions on Compact Sets. In: Barral, J., Seuret, S. (eds) Further Developments in Fractals and Related Fields. Trends in Mathematics. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8400-6_2

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