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

1 LMBP - Laboratoire de Mathématiques Blaise Pascal

Abstract : Let $K$ be a compact set in $d$ 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 $\alpha$-Hölderian functions is also discussed.

Keywords : prevalence Hausdorff dimension graphs of functions

Author: Frédéric Bayart - Yanick Heurteaux -

Source: https://hal.archives-ouvertes.fr/