The projections of fractal percolation

Stochastic Analysis Seminar Series

To study turbulence,B. Mandelbrot introduced a random fractal which is called now Mandelbrot percolation or fractal percolation. The construction is as follows: given an integer M ≥ 2 and a probability 0 < p < 1. We partition the unit square Q = [0, 1]2 into M2 congruent sub-squares and we keep each of them with probability p and throw away with probability 1−p. The squares retained are called the level one squares and in each of the level one square we repeat the same process at infinitum. What ever remains after infinitely many steps it is the random fractal called A. This may be an empty set but for p > 1/M2 with positive probability A =6 0. Even for 1/M < p < 1 the Hausdorff dimension dimH(A) > 1 almost surely, conditioned A =6 0. It was proved by J. T. Chayes, L. Chayes, and R. Durrett, that there is a critical probability1 M < pc such that for all p < pc the set A is totally disconnected, however for pc < p, we can walk within A in between the opposite sides of the square Q.What we proved with Michal Rams (Warsaw) it is as follows: Assume that p > 1/M. Then conditioned on A =6 0, for all most all realizations:·   for all lines £, the orthogonal projection of A to £ contains some intervals,·   for all points P E R2, the set of angles under which we can see some points of A from P, contains some intervals,·   for all points P E R2, the set of

distances between P and the points of A contains some intervals.This is interesting in the region 1/M < p < pc, where (conditioned on non-extinction) the random set A is a dust like set (totally disconnected),

but all of its projections contain some intervals.

Joint work with Michal Rams, IMPAN Warsaw.

 

Location:
Speaker(s):

Károly Simon (Budapest University of Technology and Economics)

Date:
Monday, March 5, 2012 - 14:15
to 15:15