On the dimension spectra of infinite iterated function systems

TL;DR

This paper explores the properties of dimension spectra in conformal iterated function systems, providing new constructions that challenge existing assumptions and raise questions about their topological and metric characteristics.

Contribution

It introduces two novel constructions: a compact perfect set not realizable as a CIFS dimension spectrum, and a similarity IFS with a zero Hausdorff dimension spectrum, addressing open questions.

Findings

A compact perfect set cannot be realized as a CIFS dimension spectrum.

A similarity IFS can have a zero Hausdorff dimension spectrum.

The results provoke new conjectures about the topological properties of IFS spectra.

Abstract

The dimension spectrum of a conformal iterated function system (CIFS) is the set of all Hausdorff dimensions of its various subsystem limit sets. This brief note provides two constructions -- (i) a compact perfect set that cannot be realized as the dimension spectrum of a CIFS; and (ii) a similarity IFS whose dimension spectrum has zero Hausdorff dimension, and thus is not uniformly perfect -- which resolve questions posed by Chousionis, Leykekhman, and Urba\'nski, and go on provoke fresh conjectures and questions regarding the topological and metric properties of IFS dimension spectra.