Multifractal spectrum and generic properties of functions monotone in several variables

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 E f h 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 ∈ [ 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.