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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Balka, Richárd Darji, Udayan B. and Elekes, Márton 2016. Hausdorff and packing dimension of fibers and graphs of prevalent continuous maps. Advances in Mathematics, Vol. 293, p. 221.


    Liu, Jia Tan, Bo and Wu, Jun 2016. Graphs of continuous functions and packing dimension. Journal of Mathematical Analysis and Applications, Vol. 435, Issue. 2, p. 1099.


    Fraser, Jonathan M. and Hyde, James T. 2015. A note on the 1-prevalence of continuous images with full Hausdorff dimension. Journal of Mathematical Analysis and Applications, Vol. 421, Issue. 2, p. 1713.


    Fraser, J. M. Orponen, T. and Sahlsten, T. 2013. On Fourier Analytic Properties of Graphs. International Mathematics Research Notices,


    KELGIANNIS, GIORGOS and LASCHOS, VAIOS 2013. ON A CONJECTURE REGARDING THE UPPER GRAPH BOX DIMENSION OF BOUNDED SUBSETS OF THE REAL LINE. Fractals, Vol. 21, Issue. 03n04, p. 1350017.


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  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 151, Issue 2
  • September 2011, pp. 355-372

The horizon problem for prevalent surfaces

  • K. J. FALCONER (a1) and J. M. FRASER (a1)
  • DOI: http://dx.doi.org/10.1017/S030500411100048X
  • Published online: 13 July 2011
Abstract

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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[2]K. J. Falconer Fractal Geometry: Mathematical Foundations and Applications (John Wiley, 2nd Ed., 2003).

[3]K. J. Falconer and M. Järvenpää Packing dimensions of sections of sets. Math. Proc. Camb. Phil. Soc. 125 (1999), 89104.

[4]K. J. Falconer and J. Lévy Véhel Horizons of fractional Brownian surfaces. Proc. Roy. Soc. Lond. 456 (2000), 21532178.

[6]B. R. Hunt , T. Sauer and J. A. Yorke Prevalence: a translational-invariant “almost every” on infinite dimensional spaces. Bull. Amer. Math. Soc. (N.S.) 27 (1992), 217238.

[9]R. D. Mauldin and S. C. Williams On the Hausdorff dimension of some graphs. Trans. Amer. Math. Soc. 298 (1986), 793803.

[11]W. Ott and J. A. Yorke Prevalence. Bull. Amer. Mat. Soc. 42 (2005), 263290.

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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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