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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 Im4 $\#956 _{}_{\#8214 \#8711 \#8214 }$ at x = 0 and at scale r in Im5 $\#8477 ^d$ is given by:

Im6 $\mstyle {T_\#936 {\#8214 \#8711 s\#8214 (0,r)}=\mfrac 1r^d\#8747 _R^d\#936 \mfenced o=( c=) \mfrac y ...  \#8214 }{(y)}=\mfrac 1r^d\#8747 _R^d\#936 _r{(y)~{\mtext d}}\#956 _{}_{\#8214 \#8711 \#8214 }{(y)}}$(2)

with wavelet $ \upper_psi$ and

Im7 $\mstyle {\#936 _r{(y)}=\#936 \mfenced o=( c=) \mfrac yr.}$

Therefore, it is important to understand, both qualitatively and quantitatively, the influence of the wavelet $ \upper_psi$ 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 r$ \rightarrow$0 , as long as $ \upper_psi$(0)$ \ne$0 , but if the wavelet $ \upper_psi$ decreases towards infinity according to a power law : Im8 ${{\#936 (x)\#8764 \#8214 x}\#8214 ^{-n_0}}$ when Im9 ${\#8214 x\#8214 \#8594 \#8734 }$ , powers of n0 pop up in the integral which may change the behaviour of Im4 $\#956 _{}_{\#8214 \#8711 \#8214 }$ '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.


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