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

TL;DR

This paper explores permeable sets in , analyzing their relation to measure and dimension, and applies these findings to characterize permeability in self-similar fractal sets.

Contribution

It introduces the concept of permeability for sets in , establishes its relation to measure and various dimensions, and characterizes permeability in self-similar fractals.

Findings

Most sets with dimension less than d-1 are permeable.

Permeability relates to Lebesgue measure and dimension notions.

Self-similar sets with certain properties are characterized as permeable.

Abstract

We study {\it permeable} sets. These are sets \(\Theta \subset \mathbb{R}^d\) which have the property that each two points \(x,y\in \mathbb{R}^d\) can be connected by a short path \(\gamma\) which has small (or even empty, apart from the end points of \(\gamma\)) intersection with \(\Theta\). 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.