On the action of multiplicative cascades on measures

TL;DR

This paper investigates how Mandelbrot multiplicative cascades transform probability measures on symbolic spaces, providing sharp criteria for measure non-degeneracy, dimension bounds, and applications to fractal measures.

Contribution

It offers new sharp criteria and bounds for the dimensions and non-degeneracy of measures under multiplicative cascades, extending previous results.

Findings

Sharp non-degeneracy criterion based on Hausdorff dimensions and entropy.

Bounds for Hausdorff and packing dimensions of the limiting measure.

Applications to dimension estimates and absolute continuity of fractal measures.

Abstract

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 random weights. We also obtain sharp bounds for the lower Hausdorff and upper packing dimensions of the limiting measure. When the original measure is a Gibbs measure associated with a potential of certain modulus of continuity (weaker than H\"older), all our results are sharp. This improves results previously obtained by Kahane and Peyri\`ere, Ben Nasr, and Fan. We exploit our results to derive dimension estimates and absolute continuity for some random fractal measures.