Dimensions of prevalent continuous functions

Abstract

Let K be a compact subset of \({\mathbb{R}^{d}}\) and write C(K) for the family of continuous functions on K. In this paper we study different fractal and multifractal dimensions of functions in C(K) that are generic in the sense of prevalence. We first prove a number of general results, namely, for arbitrary “dimension” functions \({\Delta:C(K) \to \mathbb{R}}\) satisfying various natural scaling conditions, we obtain formulas for the “dimension” Δ(f) of a prevalent function f in C(K); this is the contents of Theorems 1.1–1.3. By applying Theorems 1.1–1.3 to appropriate choices of Δ we obtain the following results: we compute the (lower and upper) local dimension of a prevalent function f in C(K); we compute the (lower or upper) Holder exponent at a point x of a prevalent function f in C(K); finally, we compute the (lower or upper) Renyi dimensions of a prevalent function f in C(K). Perhaps surprisingly, in many cases our results are very different from the corresponding results for continuous functions that are generic in the sense of Baire category.