Self-conformal sets with positive Hausdorff measure

TL;DR

This paper studies the measure-theoretic properties of quasi self-similar sets, establishing conditions under which these sets are Ahlfors regular and resolving a conjecture related to dimension drops in self-conformal sets.

Contribution

It proves that positive Hausdorff measure implies Ahlfors regularity for a broad class of self-similar sets and resolves a conjecture on dimension drops linked to overlaps.

Findings

Positive Hausdorff measure implies Ahlfors regularity.

Weak separation condition is equivalent to Ahlfors regularity in certain self-conformal sets.

Dimension drops occur precisely when exact overlaps are present.

Abstract

We investigate the Hausdorff measure and content on a class of quasi self-similar sets that include, for example, graph-directed and sub self-similar and self-conformal sets. We show that any Hausdorff measurable subset of such a set has comparable Hausdorff measure and Hausdorff content. In particular, this proves that graph-directed and sub self-conformal sets with positive Hausdorff measure are Ahlfors regular, irrespective of separation conditions. When restricting to self-conformal subsets of the real line with Hausdorff dimension strictly less than one, we additionally show that the weak separation condition is equivalent to Ahlfors regularity and its failure implies full Assouad dimension. In fact, we resolve a self-conformal extension of the dimension drop conjecture for self-conformal sets with positive Hausdorff measure by showing that its Hausdorff dimension falls below the expected value if and only if there are exact overlaps.