Independent random cascades on Galton-Watson trees

Consider an independent random cascade acting on the positive Borel measure s defined on the boundary of a Galton-Watson tree. Assuming an offspring di stribution with finite moments of all orders, J. Peyriere computed the fine scale structure of an independent random cascade on Galton-Watson trees. I n this paper we use developments in the cascade theory to relax and clarify the moment assumptions on the offspring distribution. Moreover a larger cl ass of initial measures is covered and, as a result, it is shown that it is the Holder exponent of the initial measure which is the critical parameter in the Peyriere theory.