Dimensions of graphs of prevalent continuous maps

TL;DR

This paper investigates the fractal dimensions of graphs of prevalent continuous functions on uncountable compact metric spaces, extending previous results and providing new proofs for Hausdorff and packing dimensions.

Contribution

It generalizes existing theorems on fractal dimensions of function graphs, including box, packing, and Hausdorff dimensions, for spaces with finitely many isolated points.

Findings

Lower and upper box dimensions of the graph are im_B K + d and im_B K + d.

Graph's packing dimension equals im_P K + d.

Hausdorff dimension of the graph equals im_H K + d.

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.