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arXiv · 2004.10630

Dimension spectrum of infinite self-affine iterated function systems

Natalia Jurga
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Abstract

Given an infinite iterated function system (IFS) $\mathcal{F}$, we define its dimension spectrum $D(\mathcal{F})$ to be the set of real numbers which can be realised as the dimension of some subsystem of $\mathcal{F}$. In the case where $\mathcal{F}$ 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 $\mathcal{F}$ 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 Urba\'{n}ski, Selecta 2019), we construct examples to show that $D(\mathcal{F})$ need not be compact and may contain isolated points.

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Natalia Jurga. 2020-04-22. Dimension spectrum of infinite self-affine iterated function systems. https://arxiv.org/abs/2004.10630

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