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

  • Frédéric Bayart
  • Yanick Heurteaux
Part of the Trends in Mathematics book series (TM)


Let K be a compact set in 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.


Hausdorff Dimension Fractional Brownian Motion Multifractal Analysis Hurst Parameter Weierstrass Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Balka, R., Buczolich, Z., Elekes, M.: Topological Hausdorff dimension and level sets of generic continuous functions on fractals. arXiv:1108.5578 (2011)Google Scholar
  2. 2.
    Bayart, F., Heurteaux, Y.: Multifractal analysis of the divergence of Fourier series: the extreme cases. arXiv:1110:5478, submitted (2011)Google Scholar
  3. 3.
    Christensen, J.P.R.: On sets of Haar measure zero in Abelian Polish groups. Israel J. Math. 13, 255–260 (1972)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Clausel, M., Nicolay, S.: Some prevalent results about strongly monoHölder functions. Nonlinearity 23, 2101–2116 (2010)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Falconer, K.J.: Fractal geometry: Mathematical foundations and applications. Wiley, Hoboken (2003)MATHCrossRefGoogle Scholar
  6. 6.
    Falconer, K.J., Fraser, J.M.: The horizon problem for prevalent surfaces, Math. Proc. Cambridge Philos. Soc. 151, 355–372 (2011)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Fraser, J.M., Hyde, J.T.: The Hausdorff dimension of graphs of prevalent continuous functions. arXiv:1104.2206 Real Anal. Exchange 37, 333–352 (2012)Google Scholar
  8. 8.
    Fraysse, A, Jaffard S.: How smooth is almost every function in a Sobolev space?. Rev. Mat. Iboamericana 22, 663–682 (2006)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Fraysse, A., Jaffard, S., Kahane, J.P.: Quelques propriétés génériques en analyse. (French) [Some generic properties in analysis]. C. R. Math. Acad. Sci. Paris 340, 645–651 (2005)MathSciNetMATHGoogle Scholar
  10. 10.
    Gruslys, V., Jonus̃as, J., Mijovic̀, V., Ng, O., Olsen, L., Petrykiewicz I.: Dimensions of prevalent continuous functions. Monash. Math. 166, 153–180 (2012)Google Scholar
  11. 11.
    Humke, P.D., Petruska, G.: The packing dimension of a typical continuous function is 2. Bull. Am. Math. Soc. (N.S.) 27, 345–358 (1988–89)Google Scholar
  12. 12.
    Hunt, B.R.: The Hausdorff dimension of graphs of Weierstrass functions. Proc. Am. Math. Soc. 126, 791–800 (1998)MATHCrossRefGoogle Scholar
  13. 13.
    Hyde, J.T., Laschos, V., Olsen, L., Petrykiewicz, I., Shaw, A.: On the box dimensions of graphs of typical functions. J. Math. Anal. Appl. 391, 567–581 (2012)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Mauldin, R.D., Williams, S.C.: On the Hausdorff dimension of some graphs. Trans. Am. Math. Soc. 298, 793–803 (1986)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    McClure, M.: The prevalent dimension of graphs, Real Anal. Exchange 23, 241–246 (1997)MathSciNetMATHGoogle Scholar
  16. 16.
    Olsen, L.: Fractal and multifractal dimensions of prevalent measures. Indiana Univ. Math. J. 59, 661–690 (2010)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Shaw, A.: Prevalence, M. Math Dissertation, University of St. Andrews (2010)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Laboratoire de MathématiquesClermont Université, Université Blaise PascalClermont-FerrandFrance
  2. 2.Laboratoire de MathématiquesCNRS, Complexe Scientifique des Cézeaux, UMR 6620Aubiere CedexFrance

Personalised recommendations