Selfsimilar functions can be written as the superposition of similar structures, at different scales, generated by a function g. Their expressions look like wavelet decompositions. In the case where g is regular, the multifractal formalism has been proved for the corresponding selfsimilar function, for Hölder exponents smaller than the regularity of g.

We study functions which are self-similar under the action of some nonlinear dynamical systems. We compute the exact pointwise H{ö}lder regularity, then we determine the spectrum of singularities and the Besov ``smoothness'' index, and finally we prove the multifractal formalism. The main tool in our computation is the wavelet analysis.

In this paper we prove that the conjectures of Frisch and Parisi and Arneodo et al. (called the multifractal formalism for functions) may fail for some non-homogeneous selfsimilar functions on ℝ2. In these cases, we compute the correct spectrum of singularities and we show how the multifractal formalism must be modified.