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 Urbański, Selecta 2019), we construct examples to show that D({mathcal {F}}) need not be compact and may contain isolated points.
Highlights
Let F = {Si : [0, 1]d → [0, 1]d }i∈J be a finite or countable family of Euclidean differentiable contractions whose contraction ratios are uniformly bounded above by some constant α < 1
In addition to proving the Texan conjecture, Kesseböhmer and Zhu conducted the first study of dimension spectra of conformal iterated function system (IFS), highlighting the relationship between the structure of the spectrum and the decay properties of the sequence of contraction ratios of the maps belonging to the IFS
In this paper we investigate for the first time the properties of dimension spectra in the non-conformal setting, by studying the dimension spectra of infinite self-affine IFS
Summary
}im=−01, where for any I ⊆ {1, . . . , m}, FI corresponds to the set of points in [0, 1] whose
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