Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

The horizon problem for prevalent surfaces

K. J. FALCONERa1 and J. M. FRASERa1

a1 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

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.

(Received February 04 2011)

(Online publication July 13 2011)