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Title: The horizon problem for prevalent surfaces
Authors: Falconer, Kenneth John
Fraser, Jonathan Macdonald
Keywords: QA Mathematics
Issue Date: 2011
Citation: Falconer , K J & Fraser , J M 2011 , ' The horizon problem for prevalent surfaces ' Mathematical Proceedings of the Cambridge Philosophical Society , vol 151 , no. 2 , pp. 355-372 . , 10.1017/S030500411100048X
Abstract: We investigate the box dimensions of the horizon of a fractal surface defined by a function $f \in 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 $\alpha$, for $\alpha \in [2,3)$. 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.
Version: Postprint
Description: JMF was supported by an EPSRC Doctoral Training Grant whilst undertaking this work.
Status: Peer reviewed
ISSN: 0305-0041
Type: Journal article
Rights: This is an author version of this article. The published version (c) Cambridge Philosophical Society 2011 is available at
Appears in Collections:University of St Andrews Research
Pure Mathematics Research

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