On the dimension spectra of infinite iterated function systems

Tushar Das, David Simmons

Published 2019-10-22Version 1

The dimension spectrum of an iterated function system (IFS) is the set of all Hausdorff dimensions of its various subsystem limit sets. We construct a compact perfect subset of the nonnegative reals that cannot be realized as the dimension spectrum of a conformal IFS. We also provide an example of a similarity IFS whose dimension spectrum has zero Hausdorff dimension; in particular, such a dimension spectrum is not uniformly perfect. This resolves two questions posed by Chousionis, Leykekhman, and Urba\'nski (Selecta, 2019), and provokes some new conjectures and questions regarding IFS dimension spectra.