On the Action of Multiplicative Cascades on Measures

Abstract

We consider the action of Mandelbrot multiplicative cascades on the probability measures supported on a symbolic space, especially the ergodic measures. For general probability measures, we obtain almost a sharp criterion of non-degeneracy of the limiting measure; it relies on the lower and upper Hausdorff dimensions of the measure and the entropy of the weights generating the cascade. We also obtain sharp general bounds for the lower Hausdorff and upper packing dimensions of the limiting random measure when it is non-degenerate. When the original measure is a Gibbs measure associated with a measurable potential, all our results are sharp. This improves on results previously obtained by Kahane and Peyriere, Ben Nasr, and Fan, who considered the case of Markov measures. We exploit our results on general measures to derive dimensions estimates for some random measures on Bedford-McMullen carpets, as well as absolute continuity properties, with respect to their expectation, of the projections of some random statistically self-similar measures.