Title:

Geometrical and topological properties of fractal percolation

The basic 'fractal percolation' process was first proposed by Mandelbrot in 1974 and takes the following form. Let M ≥2 and P ∈ [0,1]; we start with the unit square C0 = [0,1]2; Divide C0 into M2 equal closed squares, each of sidelength M−1 , in the natural way and retain each of these squares with probability p, or else remove it with probability 1  p. We let C1 be the union of those squares retained. The process is now repeated within each square of C1 to give a new set C2⊆C1, consisting of squares of sidelength M−2. Iterating the construction in the obvious way, we obtain a decreasing sequence of sets C0⊇ C1 ⊇ C2 ⊇ ... with limit C[sub]∞ = ∩[sub]n≥1C[sub]n. The set C[sub]∞ is an example of a random Cantor set, and is typically highly intricate in nature. It may be empty, dustlike or highly connected, depending on the value of p; percolation is said to occur if C[sub]∞ contains large connected components linking opposite sides of the unit square. In this thesis we shall investigate some of the geometrical and topological properties of C[sub]∞ that hold either almost surely (with probability 1) or with nonzero probability. In particular, the following results are established. We obtain (almost sure) lower and upper bounds on the boxcounting dimension of the 'straightest' crossings in C[sub]∞ whenever percolation occurs; we also look at the distribution of the sizes of the connected components and the probability of percolation. In the threedimensional version of the process, we establish the existence of two distinct phases of percolation, corresponding to the occurrence of paths and surfaces (or 'sheets') in the limit set, and study the limiting behaviour of the phase transition to sheet percolation as M → ∞. We also consider the results of some computer simulations of fractal percolation and present a number of generalisations of the basic process and other closely related constructions.
