Dimension spectrum of infinite self-affine iterated function systems

Abstract

Given an infinite iterated function system (IFS) F, we define its dimension spectrum D(F) to be the set of real numbers which can be realised as the dimension of some subsystem of F. In the case where FF is a conformal IFS, the properties of the dimension spectrum have been studied by several authors. In this paper we investigate for the first time the properties of the dimension spectrum when F is a non-conformal IFS. In particular, unlike dimension spectra of conformal IFS which are always compact and perfect (by a result of Chousionis, Leykekhman and Urbański, Selecta 2019), we construct examples to show that D(F) need not be compact and may contain isolated points.