Abstract
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,h λ, 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 dimensionD B of the graph byh≤2−D B≤h λ.
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Communicated by Michael F. Barnsley.
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Bedford, T. Hölder exponents and box dimension for self-affine fractal functions. Constr. Approx 5, 33–48 (1989). https://doi.org/10.1007/BF01889597
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DOI: https://doi.org/10.1007/BF01889597