Dimensions of graphs of prevalent continuous maps

Abstract

Let $K$ be an uncountable compact metric space and let $C(K,{\mathbb{R}}^d)$ denote the set of continuous maps $f:K\to {\mathbb{R}}^d$ endowed with the maximum norm. The goal of this paper is to determine various fractal dimensions of the graph of the prevalent $f\in C(K,{\mathbb{R}}^d)$.

As the main result of the paper we show that if $K$ has finitely many isolated points then the lower and upper box dimension of the graph of the prevalent $f\in C(K,{\mathbb{R}}^d)$ are ${\underline{\dim }}_{B}K+d$ and ${\overline{\dim }}_{B}K+d$, respectively. This generalizes a theorem of Gruslys, Jonušas, Mijovic, Ng, Olsen, and Petrykiewicz.

We prove that the packing dimension of the graph of the prevalent $f\in C(K,{\mathbb{R}}^d)$ is ${\dim }_{P}K+d$, generalizing a result of Balka, Darji, and Elekes.

Balka, Darji, and Elekes proved that the Hausdorff dimension of the graph of the prevalent $f\in C(K,{\mathbb{R}}^d)$ equals ${\dim }_{H}K+d$. We give a simpler proof for this statement based on a method of Fraser and Hyde.