Moments, continuity and multifractal analysis of Mandelbrot martingales

Here, a Mandelbrot measure is a statistically self-similar measure mu on th e boundary of a c-ary tree, obtained by multiplying random weights indexed by the nodes of the tree. We take a particular interest in the random varia ble Y = parallel to mu parallel to : we study the existence of finite momen ts of negative orders for Y, conditionally to Y > 0, and the continuity pro perties of Y with respect to the weights. Our results on moments make possi ble to study, with probability one, the existence of a local Holder exponen t for mu, almost everywhere with respect to another Mandelbrot measure, as well as to perform the multifractal analysis of mu, under hypotheses that a re weaker than those usually assumed.