Internal symmetry in the multifractal spectrum of fully developed turbulence

In the context of multifractal theory and She-Lévêque's model describing the statistics of intermittency in fully developed turbulence, we show that the multifractal dimensions can be simply written F(α)=1+α^*-α^*ln(α^*/2) with α^*=(2β-1-α)/lnβ=2β^p, where p is the order associated to the moment 〈ɛ_r^p〉 (with p>~0) based on the rate of energy dissipation ɛ_r and β=[(1+3/√8)^1/3+(1-3/√8)^1/3]3≈0.68. Introducing the fractal dimensions Δ_p=F(α)+α^*ln(α^*/2), this leads to the recursive relation β=(Δ_p+1-Δ_∞)/(Δ_p-Δ_∞) with Δ_∞=1. This suggests the existence of an internal symmetry in the multifractal spectrum of fully developed turbulence, which reduces considerably the number of parameters necessary to characterize intermittency statistics.