ON THE THERMODYNAMIC FORMALISM FOR MULTIFRACTAL FUNCTIONS

The thermodynamic formalism for ''multifractal'' functions phi(x) is a heuristic principle that states that the singularity spectrum f(alpha ) (defined as the Hausdorff dimension of the set S(alpha) of points wh ere phi has Holder exponent alpha) and the moment scaling exponent tau (q) (giving the power law behavior of integral\phi(x + t) - phi(x)\q d x for small \t\) should be related by the Legendre transform, tau(q) = 1 + inf/alpha greater-than-or-equal-to 0 [qalpha - f(alpha)]. The ran ge of validity of this heuristic principle is unknown. Here this princ iple is rigorously verified for a family of ''toy examples'' that are solutions of refinement equations. These example functions exhibit osc illations on all scales, and correspond to multifractal signed measure s rather than multifractal measures; moreover, their singularity spect ra f(alpha) are not concave.