Furstenberg sumset conjecture and Mandelbrot percolations

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.