New methods to bound the critical probability in fractal percolation

The following full text is a preprint version which may differ from the publisher's version. Abstract: We study the critical probability p c (M) in two-dimensional M-adic fractal percolation. To find lower bounds, we compare fractal perco-lation with site percolation. Fundamentally new is the construction of an computable increasing sequence that converges to p c (M). We prove that p c (2) > 0.881 and p c (3) > 0.784. For the upper bounds, we introduce an iterative random process on a finite alphabet A , which is easier to analyze than the original process. We show that p c (2) < 0.993, p c (3) < 0.940 and p c (4) < 0.972.