We introduce multifractal pressure and dynamical multifractal zeta-functions providing precise information of a very general class of multifractal spectra, including, for example, the fine multifractal spectra of graph-directed self-conformal measures and the fine multifractal spectra of ergodic Birkhoff averages of continuous functions on graph-directed self-conformal sets.2000 Mathematics Subject Classification. Primary: 28A78. Secondary: 37D30, 37A45.

Let K be a compact subset of 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 : C(K ) → R satisfying various natural scaling conditions, we obtain formulas for the "dimension"( 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 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) Hölder exponent at a point x of a prevalent Communicated by P. Friz.

We introduce multifractal zeta-functions providing precise information of a very general class of multifractal spectra, including, for example, the multifractal spectra of self-conformal measures and the multifractal spectra of ergodic Birkhoff averages of continuous functions. More precisely, we prove that these and more general multifractal spectra equal the abscissae of convergence of the associated zetafunctions.2000 Mathematics Subject Classification. Primary: 28A78. Secondary: 37D30, 37A45., but to find explicit expressions for the quantities M µ,δ (q) and N µ,δ (α; r) themselves. The purpose of this work can bee seen as a first step in this direction. Again, summarizing this somewhat more succinctly, the present work is concentrated on the following problem: Present work:This work explores methods of finding explicit expressions for M µ,δ (q) and N µ,δ (α; r) . It is clear that finding explicit expressions for M µ,δ (q) and N µ,δ (α; r) is a more challenging undertaking than determining the limiting behaviour of the ratios log M µ,δ (q) − log δ and log N µ,δ (α;r) − log δ ; indeed, if explicit expressions for M µ,δ (q) and N µ,δ (α; r) are known, then the limiting behaviour of the ratios log M µ,δ (q) − log δ and log N µ,δ (α;r) − log δ

No abstract