Measure and dimension theory of permeable sets and its applications to fractals

TL;DR

This paper develops a comprehensive framework for the permeability of subsets of \\mathbb{R}^d, linking geometric path-intersections to measure and dimension theory. It introduces null, finite, and standard permeability, and establishes norm-invariance results that unify when the unit ball is strictly convex. It then maps permeability onto Lebesgue measure and a spectrum of dimension notions, showing that many dimensions (including Hausdorff, box, and Assouad) do not uniquely determine permeability, while Nagata dimension provides a robust null-permeability criterion, especially for self-similar sets. The self-similar and self-affine sections culminate in general permeability criteria: planar sets with finite type are permeable, higher-dimensional sets with finite type are null permeable, and explicit impermeable Bedford-McMullen carpets demonstrate sharp boundaries, highlighting the nuanced interplay between fractal structure and short-path connectivity. Overall, the work connects geometric measure theory, dimension theory, and fractal geometry to yield practical criteria for assessing the Lipschitz properties of maps in the presence of complicated sets.

Abstract

We study {\it permeable} sets. These are sets \(Θ\subset \mathbb{R}^d\) which have the property that each two points \(x,y\in \mathbb{R}^d\) can be connected by a short path \(γ\) which has small (or even empty, apart from the end points of \(γ\)) intersection with \(Θ\). We investigate relations between permeability and Lebesgue measure and establish theorems on the relation of permeability with several notions of dimension. It turns out that for most notions of dimension each subset of \(\mathbb{R}^d\) of dimension less than \(d-1\) is permeable. We use our permeability result on the Nagata dimension to characterize permeability properties of self-similar sets with certain finiteness properties.

Figures (15)

• Figure 1: Connecting points by short paths. (A) If we allow for small detours, we can connect each pair of points by intersecting the line in at most one point. (B) For the strip we get an uncountable intersection even with small detours. (C) Can we always get countable intersections with the Sierpiński triangle with small detours?
• Figure 2: Two approximations of the set $\Theta$. The width of the gaps between the squares tends to zero much faster than the width of the squares. We construct a path $\gamma$ connecting two points $(x_1,y_1),(x_2,y_2)\in {\mathbb R}^2$ that consists of axis-parallel line segments inside the gaps. This path $\gamma$ has countable intersection with $\Theta$.
• Figure 3: One, two, and ten iterations of Osgood's construction
• Figure 4: One, two, and ten iterations of Sierpiński's construction
• Figure 5: Five iterations of a permeable Osgood curve

Theorems & Definitions (102)

• Definition 1.1: Length of a path
• Definition 1.2: Permeability
• Remark 1.4: Relation to percolation
• Lemma 1.5
• proof
• Lemma 1.6
• proof
• Theorem 1.7
• Theorem 1.8
• Lemma 1.9