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

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

2011-12-08

arxiv.org/abs/1112.1931v1

Mathematics

Classical Analysis and ODEs

Scientific paper

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.

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