Multifractal spectrum and generic properties of functions monotone in several variables
Résumé
We study the singularity (multifractal) spectrum of continuous functions monotone in several variables. We find an upper bound valid for all functions of this type, and we prove that this upper bound is reached for generic functions monotone in several variables. Let Ell be the set of points at which f has a pointwise exponent equal to h. For generic monotone functions f : [0, 1](d) -> R, we have that dim E (f)(h)= d - 1 + h for all h is an element of [0,1], and in addition, we obtain that the set E(f)(h) is empty as soon as h > 1. We also investigate the level set structure of such functions. (C) 2011 Elsevier Inc. All rights reserved.