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

Résumé

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 $\alpha$-Hölderian functions is also discussed.
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Dates et versions

hal-00649590 , version 1 (08-12-2011)

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Frédéric Bayart, Yanick Heurteaux. On the Hausdorff dimension of graphs of prevalent continuous functions on compact sets. Fractals and Related Fields 2, Jun 2011, Porquerolles Island, France. pp.25-34. ⟨hal-00649590⟩
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