Quantifying inhomogeneity in fractal sets

TL;DR

This paper develops a quantitative framework to measure and analyze the inhomogeneity of fractal sets, using Assouad dimension and large deviations principles, with applications to specific invariant sets.

Contribution

It introduces a new quantitative theory of inhomogeneity in fractals, linking Assouad dimension differences to measure-based inhomogeneity analysis and large deviations.

Findings

The measure of inhomogeneity satisfies a Large Deviations Principle.

Rate functions provide insights into the inhomogeneity of different fractal sets.

Comparison of invariant sets reveals the depth of their inhomogeneity.

Abstract

An inhomogeneous fractal set is one which exhibits different scaling behaviour at different points. The Assouad dimension of a set is a quantity which finds the `most difficult location and scale' at which to cover the set and its difference from box dimension can be thought of as a first-level overall measure of how inhomogeneous the set is. For the next level of analysis, we develop a quantitative theory of inhomogeneity by considering the measure of the set of points around which the set exhibits a given level of inhomogeneity at a certain scale. For a set of examples, a family of $(\times m, \times n)$-invariant subsets of the 2-torus, we show that this quantity satisfies a Large Deviations Principle. We compare members of this family, demonstrating how the rate function gives us a deeper understanding of their inhomogeneity.