Dimension Conservation for Self-Similar Sets and Fractal Percolation

Abstract

We introduce a technique that uses projection properties of fractal percolation to establish dimension conservation results for sections of deterministic self-similar sets. For example, let K be a self-similar subset of R 2 with Hausdorff dimension dim H K > 1 such that the rotational components of the underlying similarities generate the full rotation group. Then for all > 0, writing π θ for projection onto the line L θ in direction θ, the Hausdorff dimensions of the sections satisfy dim H (K ∩ π −1 θ x) > dim H K − 1 − for a set of x ∈ L θ of positive Lebesgue measure, for all directions θ except for those in a set of Hausdorff dimension 0. For a class of self-similar sets we obtain a similar conclusion for all directions, but with lower box dimension replacing Hausdorff dimensions of sections. We obtain similar inequalities for the dimensions of sections of Mandelbrot percolation sets.

A self-similar measure with dense rotations, singular projections and discrete slices

“…Another related result for sets is due to Falconer and Jin [FJ2]. They showed that if K ⊂ R 2 is self-similar, with dim K > 1 and a dense rotation group, then for every…”

Fractal percolations

“…Note that the authors of [6] claim dimension conservation almost surely for any fixed direction, but not almost surely for all of them simultaneously. In a following paper [7] the same authors obtain dimension conservation for all directional projections at once in the sense of (3.2). Since the proof is beautiful, we sketch its idea below:…”

“…In the last years several authors received dimension conservation results on projections of fractal percolation sets [6,7,22]. First let us introduce three dimension conservation notions relevant for this paper.…”

“…The authors of [7] use this for a typical (conditioned on non-extinction) realization A := E ω (p). Namely, by (3.6), the condition (3.8) holds for all α ∈ A d ,k .…”

On self-similar measures with absolutely continuous projections and dimension conservation in each direction

“…An analogues statement, for self-similar sets with the SSC, was first proven by Furstenberg [Fur]. Another related result for sets is due to Falconer and Jin [FJ2]. They showed that if K ⊂ C is self-similar, with dim K > 1 and a dense rotation group, then for every ǫ > 0 there exists N ǫ ⊂ S, with dim H N ǫ = 0, such that for z ∈ S \ N ǫ the set {x ∈ R : dim H (K ∩ P −1 z {x}) > dim K − 1 − ǫ} has positive length.…”