Hölder exponents and box dimension for self-affine fractal functions Authors Tim Bedford King's College Research Centre King's College Article

Received: 15 December 1986 DOI :
10.1007/BF01889597

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

AMS classification
26A30
41A30
58F11

Key words and phrases
Fractals
Self-affine
Hölder exponents
Box dimension
Gibbs measures
Communicated by Michael F. Barnsley.

References [Ba]

M. F. Barnsley (1985): Fractal Functions and Interpolation. Atlanta: Georgia Institute of Technology Preprints.

[BH]

M. F. Barnsley, A. N. Harrington (1985): The Calculus of Fractal Interpolation Functions. Atlanta: Georgia Institute of Technology Preprints.

[Be]

T. J. Bedford (1986):Dimension and dynamics for fractal recurrent sets . J. London Math. Soc. (2),33 :89–100.

[BU]

A. S. Besicovitch, H. D. Ursell (1937):Sets of fractional dimensions, V: On dimensional numbers of some continuous curves . J. London Math. Soc.,12 :18–25.

[Bo1]

R. Bowen (1975): Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Mathematics, Vol. 470. Berlin: Springer-Verlag.

[Bo2]

R. Bowen (1979): Hausdorff Dimension of Quasi-Circles. Publications Mathématiques, Vol. 50. Paris: Institut des Hautes Etudes Scientifiques, pp. 11–25.

[Ca]

H. Cajar (1981): Billingsley Dimension in Probability Spaces. Lecture Notes in Mathematics, Vol. 892. Berlin: Springer-Verlag.

[FHY]

A. Fathi, M. R. Herman, J. C. Yoccoz (1981):A proof of Pesin's stable manifold theorem . In: Geometric Dynamics (J. Palis Jr., ed.). Lecture Notes in Mathematics, Vol. 1007. Berlin, Springer-Verlag.

[Ha]

D. P.Hardin (1986): Personal communication.

[HM]

D. P. Hardin, P. R. Massopust (1986):The capacity for a class of functions . Comm. Math. Phys.,105 :455–460.

[Hu]

J. E. Hutchinson (1981):Fractals and self-similarity . Indiana Univ. Math. J.,30 :713–747.

[Tr]

C.Tricot :Two definitions of fractional dimension . Math. Proc. Cambridge Philos. Soc.,91 :57–74.

© Springer-Verlag New York Inc 1989