On the action of multiplicative cascades on measures

We consider the action of Mandelbrot multiplicative cascades on probability measures supported on a symbolic space. 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 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. We exploit our
results to derive dimension estimates and absolute continuity for some random fractal measures.