Minkowski Content and local Minkowski Content for a class of self-conformal sets

TL;DR

This paper studies how Minkowski measurability of self-similar sets is affected by smooth transformations, providing formulas for Minkowski content and showing that the property is not always preserved.

Contribution

It establishes the relationship between Minkowski measurability of self-similar sets and their smooth images, including explicit formulas and counterexamples.

Findings

Minkowski measurability of $K$ implies measurability of $F$ with explicit content formula.

Counterexample shows $F$ can be measurable even if $K$ is not.

Average Minkowski content exists for both $K$ and $F$ with explicit relation.

Abstract

We investigate (local) Minkowski measurability of $\mathcal C^{1+\alpha}$ images of self-similar sets. We show that (local) Minkowski measurability of a self-similar set $K$ implies (local) Minkowski measurability of its image $F$ and provide an explicit formula for the (local) Minkowski content of $F$ in this case. A counterexample is presented which shows that the converse is not necessarily true. That is, $F$ can be Minkowski measurable although $K$ is not. However, we obtain that an average version of the (local) Minkowski content of both $K$ and $F$ always exists and also provide an explicit formula for the relation between the (local) average Minkowski contents of $K$ and $F$.