Weakly controlled Moran constructions and iterated functions systems in metric spaces

TL;DR

This paper investigates the Hausdorff measures of limit sets generated by weakly controlled Moran constructions and semiconformal iterated function systems in metric spaces, generalizing classical results.

Contribution

It introduces new separation conditions for semiconformal iterated function systems and extends known results to broader metric space settings.

Findings

Separation conditions influence Hausdorff measure of limit sets

Generalization of self-similar set results to metric spaces

Extension of controlled Moran construction results

Abstract

We study the Hausdorff measures of limit sets of weakly controlled Moran constructions in metric spaces. The separation of the construction pieces is closely related to the Hausdorff measure of the corresponding limit set. In particular, we investigate different separation conditions for semiconformal iterated function systems. Our work generalizes well known results on self-similar sets in metric spaces as well as results on controlled Moran constructions in Euclidean spaces.