## Section: New Results

Keywords : microcanonical multifractal formalism, wavelets, singularity exponents.

### Limiting behaviour of wavelets and singularity exponents

Participants : Hussein Yahia, Antonio Turiel [ ICM ] .

The accurate computation of singularity exponents is a key aspect of the Microcanonical Multifractal Formalism (MMF). The singularity exponents are computed using a wavelet projection of a multifractal measure: the wavelet projection of measure at x = 0 and at scale r in is given by:

with wavelet and

Therefore, it is important to understand, both qualitatively and quantitatively, the influence of the wavelet on the computation of the singularity exponent at x = 0 .

We study the influence of the wavelet's tail at infinity on the
singularity exponent computed at x = 0 . Using the Lebesgue
density theorem and the analysis of the integral appearing
in (2 ), we obtain that the local behaviour
of the wavelet at x = 0 lets the singularity exponent at
0 unchanged when r0 , as long as
(0)0 , but if the wavelet decreases towards
infinity according to a power law : when , powers of n_{0}
pop up in the integral which may change the behaviour of
's singularity exponent at 0 , when r tend towards 0. This qualitative analysis is the first step
towards a mathematical understanding of numerical artefacts appearing
in the algorithms implementing the wavelet projection. The second step
will consist in computing the quantitative dependance of the wavelet's
tail on the singularity exponents.