OSCILLATING SINGULARITIES ON CANTOR SETS - A GRAND-CANONICAL MULTIFRACTAL FORMALISM

The singular behavior of functions is generally characterized by their Holder exponent. However, we show that this exponent poorly character izes oscillating singularities. We thus introduce a second exponent th at accounts for the oscillations of a singular behavior and we give a characterization of this exponent using the wavelet transform. We then elaborate on a ''grand-canonical'' multifractal formalism that descri bes statistically the fluctuations of both the Holder and the oscillat ion exponents. We prove that this formalism allows us to recover the g eneralized singularity spectrum of a large class of fractal functions involving oscillating singularities.