NO DIRECTED FRACTAL PERCOLATION IN ZERO AREA

We consider the fractal percolation process on the unit square with fi xed decimation parameter N and level-dependent retention parameters {p (k)}; that is, for all k greater than or equal to 1, at the kth stage every retained square of side length N1-k is partitioned into N-2 cong ruent subsquares, and each of these is retained with probability p(k). independent of all others. We show that if Pi(k) p(k) = 0 (i.e., if t he area of the limiting set vanishes a.s.), then a.s. the limiting set contains no directed crossings of the unit square (a directed crossin g is a path that crosses the unit square from left to right, and moves only up, down, and to the right).