Multiscaling transformation in dynamical systems and turbulence

Multiscaling is a scaling law where the exponent is slowing varying with the length scale (pseudo-algebraic law). We discuss its origin as a consequences of multifractility and the existence of a lower cutoff in the calculation of correlation functions in different contexts. We derive some exact results in the case of two scale Cantor sets, which can be extended to other fractal structures such as strange attractors of chaotic systems. In fully developed turbulence, the cutoff is naturally introduced by the viscosity and our approach leads to the prediction of an intermediate dissipation range, which can be tested experimentally.