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The horizon problem for prevalent surfaces

  • K. J. FALCONER (a1) and J. M. FRASER (a1)

We investigate the box dimensions of the horizon of a fractal surface defined by a function fC[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.

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[1]Falconer K. J. The horizon problem for random surfaces. Math. Proc. Camb. Phil. Soc. 109 (1991), 211219.
[2]Falconer K. J. Fractal Geometry: Mathematical Foundations and Applications (John Wiley, 2nd Ed., 2003).
[3]Falconer K. J. and Järvenpää M. Packing dimensions of sections of sets. Math. Proc. Camb. Phil. Soc. 125 (1999), 89104.
[4]Falconer K. J. and Lévy Véhel J. Horizons of fractional Brownian surfaces. Proc. Roy. Soc. Lond. 456 (2000), 21532178.
[5]Gruslys V., Jonušas J., Mijovic V., Ng O., Olsen L. and Petrykiewicz I. Dimensions of prevalent continuous functions. preprint (2010).
[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), 217238.
[7]Hyde J. T., Laschos V., Olsen L., Petrykiewicz I. and Shaw A. On the box dimensions of graphs of typical functions. preprint (2010).
[8]Massopust P. R. Fractal Functions, Fractal Surfaces, and Wavelets (Academic Press, 1994).
[9]Mauldin R. D. and Williams S. C. On the Hausdorff dimension of some graphs. Trans. Amer. Math. Soc. 298 (1986), 793803.
[10]McClure M. The prevalent dimension of graphs. Real Anal. Exchange 23 (1997), 241246.
[11]Ott W. and Yorke J. A. Prevalence. Bull. Amer. Mat. Soc. 42 (2005), 263290.
[12]Rudin W. Functional Analysis (McGraw-Hill, 2nd Ed., 1991).
[13]Shaw A. Prevalence. M. Math Dissertation (2010).
[14]Wingren P. Dimensions of graphs of functions and lacunary decompositions of spline approximations. Real Anal. Exchange 26 (2000), 1726.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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