ON THE THERMODYNAMIC FORMALISM FOR MULTIFRACTAL FUNCTIONS

Abstract

The thermodynamic formalism for "multifractal" functions φ(x) is a heuristic principle that states that the singularity spectrum f(α) (defined as the Hausdorff dimension of the set Sα of points where φ has Hölder exponent α) and the moment scaling exponent τ(q) (giving the power law behavior of ∫ |φ(x + t) – φ (x)|q dx for small |t|) should be related by the Legendre transform, [Formula: see text]. The range of validity of this heuristic principle is unknown. Here this principle is rigorously verified for a family of "toy examples" that are solutions of refinement equations. These example functions exhibit oscillations on all scales, and correspond to multifractal signed measures rather than multifractal measures; moreover, their singularity spectra f(α) are not concave.