Furstenberg sumset conjecture and Mandelbrot percolations

TL;DR

This paper extends key theorems in fractal geometry and measure theory to new classes of measures, including Mandelbrot cascades and percolations, revealing new dimension results and generalizations.

Contribution

It generalizes Hochman and Shmerkin's projection theorem to product measures of Mandelbrot cascades and extends Furstenberg's sumset theorem to these measures and their images.

Findings

Extended projection theorem to Mandelbrot cascades

Generalized Furstenberg sumset theorem to new measure classes

Derived dimension results for convolutions of Bernoulli and Mandelbrot measures

Abstract

In this paper we extend Hochman and Shmerkin's projection theorem to product measures of Mandelbrot cascades acting on ergodic measures imaged through canonical mappings of one-dimensional iterated function systems without any separation conditions. Consequently we extend Furstenberg sumset theorem to images of subshifts on symbolic spaces, and to Mandelbrot percolations on invariant sets. We also obtain dimension results for convolutions of Bernoulli convolutions and that of Mandelbrot cascade measures.