Mandelbrot multiplicative cascades provide a construction of a dynamical system on a set of probability measures de ned by inequalities on moments. To be more speci c, beyond the rst iteration, the trajectories take values in the set of xed points of smoothing transformations (i.e., some generalized stable laws). Studying this system leads to a central limit theorem and to its functional version. The limit Gaussian process can also be obtained as limit of an 'additive cascade' of independent normal variables.