On the dimension spectra of infinite iterated function systems

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 Urbański, and go on provoke fresh conjectures and questions regarding the topological and metric properties of IFS dimension spectra.

References in corpus (2)

• Rigorous effective bounds on the Hausdorff dimension of continued fraction Cantor sets: a hundred decimal digits for the dimension of $E_2$
• Dimension rigidity in conformal structures