Constructive Approximation

, Volume 5, Issue 1, pp 33–48

Hölder exponents and box dimension for self-affine fractal functions

  • Tim Bedford

DOI: 10.1007/BF01889597

Cite this article as:
Bedford, T. Constr. Approx (1989) 5: 33. doi:10.1007/BF01889597


We consider some self-affine fractal functions previously studied by Barnsleyet al. The graphs of these functions are invariant under certain affine scalings, and we extend their definition to allow the use of nonlinear scalings. The Hölder exponent,h, for these fractal functions is calculated and we show that there is a larger Hölder exponent,hλ, defined at almost every point (with respect to Lebesgue measure). For a class of such functions defined using linear affinities these exponents are related to the box dimensionDB of the graph byh≤2−DBhλ.

AMS classification

26A30 41A30 58F11 

Key words and phrases

Fractals Self-affine Hölder exponents Box dimension Gibbs measures 

Copyright information

© Springer-Verlag New York Inc 1989

Authors and Affiliations

  • Tim Bedford
    • 1
  1. 1.King's College Research CentreKing's CollegeCambridgeEngland
  2. 2.Department of Mathematics and InformaticsDelft University of TechnologyDelftThe Netherlands

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