On the Minkowski content of self-similar random homogeneous iterated function systems

TL;DR

This paper investigates the Minkowski content of random homogeneous self-similar sets, proving that certain measures are almost surely infinite or zero, contrasting with previous models and expectations.

Contribution

It demonstrates that the upper Minkowski content and upper average Minkowski content are almost surely infinite for these sets, resolving a conjecture and highlighting stark differences from other models.

Findings

Upper Minkowski content is almost surely infinite.

Lower Minkowski content is zero in the equicontractive case.

Lower average Minkowski content is infinite.

Abstract

The Minkowski content of a compact set is a fine measure of its geometric scaling. For Lebesgue null sets it measures the decay of the Lebesgue measure of epsilon neighbourhoods of the set. It is well known that self-similar sets, satisfying reasonable separation conditions and non-log comensurable contraction ratios, have a well-defined Minkowski content. When dropping the contraction conditions, the more general notion of average Minkowski content still exists. For random recursive self-similar sets the Minkowski content also exists almost surely, whereas for random homogeneous self-similar sets it was recently shown by Z\"{a}hle that the Minkowski content exists in expectation. In this short note we show that the upper Minkowski content, as well as the upper average Minkowski content of random homogeneous self-similar sets is infinite, almost surely, answering a conjecture posed by Z\"{a}hle. Additionally, we show that in the random homogeneous equicontractive self-similar setting the lower Minkowski content is zero and the lower average Minkowski content is also infinite. These results are in stark contrast to the random recursive model or the mean behaviour of random homogeneous attractors.