On the Hausdorff dimension of graphs of prevalent continuous functions on compact sets

TL;DR

This paper proves that for a prevalent set of continuous functions on a compact set with positive Hausdorff dimension, the graph's Hausdorff dimension reaches its maximum possible value, extending previous results and including H"olderian functions.

Contribution

It generalizes prior work by showing the maximal Hausdorff dimension of graphs for prevalent continuous functions on compact sets, using Fractional Brownian Motion.

Findings

Hausdorff dimension of graphs equals dim_H(K)+1 for prevalent functions

Results extend to ta-Hf6lderian functions

Generalizes previous work by Fraser and Hyde

Abstract

Let $K$ be a compact set in $\rd$ with positive Hausdorff dimension. Using a Fractional Brownian Motion, we prove that in a prevalent set of continuous functions on $K$, the Hausdorff dimension of the graph is equal to $\dim_{\mathcal H}(K)+1$. This is the largest possible value. This result generalizes a previous work due to J.M. Fraser and J.T. Hyde which was exposed in the conference {\it Fractal and Related Fields~2}. The case of $\alpha$-H\"olderian functions is also discussed.