The Geometry of Fractal Percolation

Abstract

Abstract. A well studied family of random fractals called fractal percolation is discussed. We focus on the projections of fractal percolation on the plane. Our goal is to present stronger versions of the classical Marstrand theorem, valid for almost every realization of fractal percolation. The extensions go in three directions:• the statements work for all directions, not almost all,• the statements are true for more general projections, for example radial projections onto a circle, • in the case dim H > 1, each projection has not only positive Lebesgue measure but also has nonempty interior. introductionTo model turbulence, Mandelbrot [13, 14] introduced a statistically self-similar family of random Cantor sets. Since that time this family has got at least three names in the literature: fractal percolation, Mandelbrot percolation and canonical curdling, among which we will use the first one. In 1996 Lincoln Chayes [3] published an excellent survey giving an account about the most important results known in that time. His survey focused on the percolation related properties while we place emphasis on the geometric measure theoretical properties (projections and slices) of fractal percolation sets. About the projections of a general Borel set the celebrated Marstrand Theorem gives the following information:2000 Mathematics Subject Classification. Primary 28A80 Secondary 60J80, 60J85Key words and phrases. Random fractals, Hausdorff dimension, processes in random environment.

Dimension Conservation for Self-Similar Sets and Fractal Percolation

“…There has been great interest recently in identifying classes of sets, in particular classes of self-similar sets and their variants, for which these various inequalities hold for all, rather than just almost all, subspaces. Several papers establish (1.1) for all projections for classes of self-similar sets [8,10,21,25] and for percolation on self-similar sets [7,22,23,24,26]. Here we consider dimensions of sections, and identify sets for which (1.4), or a similar inequality for box-counting dimension, holds for all subspaces V .…”

Projections of planar Mandelbrot random measures

“…, µ 0 -almost everywhere. Moreover, we deduce from Theorem 3.2(2) and the proof of Proposition 8.12 that for π * µ 0 -almost every x, for µ x 0 -almost every y, we 56 have both…”

Self-Similar Sets: Projections, Sections and Percolation

“…The topological properties of Mandelbrot percolation have been extensively studied, see [3,6,34]. In particular, there is a critical probability p c with 1/M < p c < 1 such that if p > p c then, conditional on non-extinction, E p contains many connected components, so its projections onto all lines necessarily have positive Lebesgue measure.…”

“…If p ≤ p c the percolation set E p is totally disconnected, and Marstrand's theorems provide information on its projections in almost all directions. However, Rams and Simon [33,34,35] recently showed using a careful geometrical analysis that, conditional on E p = / 0, almost surely the conclusions of Theorem 1.1 hold for all projections simultaneously.…”

“…For example, the conclusions of Theorem 1.1 hold for all projections for certain classes of self-similar sets [10,15,31,36] as well as for random subsets of certain self-similar sets [7,33,35,34,38].…”