The horizon problem for prevalent surfaces

TL;DR

This paper explores the box dimensions of horizons of fractal surfaces, showing prevalent surfaces satisfy the horizon property, and investigates the behavior in spaces with surfaces of various dimensions, revealing nuanced results.

Contribution

It introduces a framework for analyzing the horizon property in fractal surfaces of different dimensions and demonstrates the property holds in Lipschitz spaces but not universally.

Findings

Prevalent surfaces of box dimension 3 satisfy the horizon property.

In spaces with surfaces of dimension between 2 and 3, the horizon's lower box dimension varies.

The horizon property holds in Lipschitz spaces for prevalent functions.

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.