Oscillating singularities on cantor sets: A grand-canonical multifractal formalism

The singular behavior of functions is generally characterized by their Hölder exponent. However, we show that this exponent poorly characterizes oscillating singularities. We thus introduce a second exponent that 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 describes statistically the fluctuations of both the Hölder and the oscillation exponents. We prove that this formalism allows us to recover the generalized singularity spectrum of a large class of fractal functions involving oscillating singularities.