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The Hausdorff dimension of graphs of prevalent continuous functions

By Jonathan M. Fraser and James T. Hyde

Abstract

We prove that the Hausdorff dimension of the graph of a prevalent continuous function is 2. We also indicate how our results can be extended to the space of continuous functions on $[0,1]^d$ for $d \in \mathbb{N}$ and use this to obtain results on the `horizon problem' for fractal surfaces. We begin with a survey of previous results on the dimension of a generic continuous function

Topics: Mathematics - Metric Geometry, Mathematics - Probability, 28A80, 28A78
Year: 2011
OAI identifier: oai:arXiv.org:1104.2206
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