Dimension spectrum of infinite self-affine iterated function systems

Abstract

AbstractGiven an infinite iterated function system (IFS) $${\mathcal {F}}$$ F , we define its dimension spectrum $$D({\mathcal {F}})$$ D ( F ) to be the set of real numbers which can be realised as the dimension of some subsystem of $${\mathcal {F}}$$ F . In the case where $${\mathcal {F}}$$ F 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 $${\mathcal {F}}$$ 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 $$D({\mathcal {F}})$$ D ( F ) need not be compact and may contain isolated points.