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 $f\colon K \to \mathbb{R}^d$ 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, Jonu\v{s}as, Mijovi\`c, Ng, Olsen, and Petrykiewicz. We prove that the graph of the prevalent $f\in C(K,\mathbb{R}^d)$ has packing dimension $\dim_P K+d$, 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.