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The horizon problem for prevalent surfaces

Published online by Cambridge University Press:  13 July 2011

K. J. FALCONER  and
J. M. FRASER
K. J. FALCONER
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, UK e-mail: kjf@st-and.ac.uk and jmf32@st-and.ac.uk
J. M. FRASER
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, UK e-mail: kjf@st-and.ac.uk and jmf32@st-and.ac.uk
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Abstract

We investigate the box dimensions of the horizon of a fractal surface defined by a function f ∈ C[0,1]2. In particular we show that a prevalent surface satisfies the ‘horizon property’, namely that the box dimension of the horizon is one less than that of the surface. Since a prevalent surface has box dimension 3, this does not give us any information about the horizon of surfaces of dimension strictly less than 3. To examine this situation we introduce spaces of functions with surfaces of upper box dimension at most α, for α ∈ [2,3). In this setting the behaviour of the horizon is more subtle. We construct a prevalent subset of these spaces where the lower box dimension of the horizon lies between the dimension of the surface minus one and 2. We show that in the sense of prevalence these bounds are as tight as possible if the spaces are defined purely in terms of dimension. However, if we work in Lipschitz spaces, the horizon property does indeed hold for prevalent functions. Along the way, we obtain a range of properties of box dimensions of sums of functions.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 151 , Issue 2 , September 2011 , pp. 355 - 372
Copyright
Copyright © Cambridge Philosophical Society 2011

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References

REFERENCES

[1]Falconer, K. J.The horizon problem for random surfaces. Math. Proc. Camb. Phil. Soc. 109 (1991), 211–219.CrossRefGoogle Scholar
[2]Falconer, K. J.Fractal Geometry: Mathematical Foundations and Applications (John Wiley, 2nd Ed., 2003).CrossRefGoogle Scholar
[3]Falconer, K. J. and Järvenpää, M.Packing dimensions of sections of sets. Math. Proc. Camb. Phil. Soc. 125 (1999), 89–104.CrossRefGoogle Scholar
[4]Falconer, K. J. and Lévy Véhel, J.Horizons of fractional Brownian surfaces. Proc. Roy. Soc. Lond. 456 (2000), 2153–2178.CrossRefGoogle Scholar
[5]Gruslys, V., Jonušas, J., Mijovic, V., Ng, O., Olsen, L. and Petrykiewicz, I. Dimensions of prevalent continuous functions. preprint (2010).CrossRefGoogle Scholar
[6]Hunt, B. R., Sauer, T. and Yorke, J. A.Prevalence: a translational-invariant “almost every” on infinite dimensional spaces. Bull. Amer. Math. Soc. (N.S.) 27 (1992), 217–238.CrossRefGoogle Scholar
[7]Hyde, J. T., Laschos, V., Olsen, L., Petrykiewicz, I. and Shaw, A. On the box dimensions of graphs of typical functions. preprint (2010).Google Scholar
[8]Massopust, P. R.Fractal Functions, Fractal Surfaces, and Wavelets (Academic Press, 1994).Google Scholar
[9]Mauldin, R. D. and Williams, S. C.On the Hausdorff dimension of some graphs. Trans. Amer. Math. Soc. 298 (1986), 793–803.CrossRefGoogle Scholar
[10]McClure, M.The prevalent dimension of graphs. Real Anal. Exchange 23 (1997), 241–246.CrossRefGoogle Scholar
[11]Ott, W. and Yorke, J. A.Prevalence. Bull. Amer. Mat. Soc. 42 (2005), 263–290.CrossRefGoogle Scholar
[12]Rudin, W.Functional Analysis (McGraw-Hill, 2nd Ed., 1991).Google Scholar
[13]Shaw, A. Prevalence. M. Math Dissertation (2010).Google Scholar
[14]Wingren, P.Dimensions of graphs of functions and lacunary decompositions of spline approximations. Real Anal. Exchange 26 (2000), 17–26.CrossRefGoogle Scholar