Dimension spectrum of infinite self-affine iterated function systems

Natalia Anna Jurga

doi:10.1007/s00029-021-00674-x

Dimension spectrum of infinite self-affine iterated function systems

Natalia Anna Jurga

Pure Mathematics

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Article number	49
Number of pages	23
Journal	Selecta Mathematica: New Series
Volume	27
Issue number	3
DOIs	https://doi.org/10.1007/s00029-021-00674-x
Publication status	Published - 16 Jun 2021

Keywords

Iterated function system
Self-affline set
Dimension spectrum
Hausdorff dimension

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10.1007/s00029-021-00674-xLicence: CC BY

Jurga_2021_SM_Dimension-Spectrum_CC
Copyright © The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
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Fourier analytic techniques: Fourier analytic techniques in geometry and analysis
Fraser, J. (PI) & Falconer, K. J. (CoI)
EPSRC
1/02/18 → 11/06/21
Project: Standard

Cite this

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title = "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 Urba{\'n}ski, Selecta 2019), we construct examples to show that D(F) need not be compact and may contain isolated points.",

keywords = "Iterated function system, Self-affline set, Dimension spectrum, Hausdorff dimension",

author = "Jurga, {Natalia Anna}",

note = "Funding: The author was financially supported by the Leverhulme Trust (Research Project Grant number RPG-2016-194) and by the EPSRC (Standard Grant EP/R015104/1).",

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AU - Jurga, Natalia Anna

N1 - Funding: The author was financially supported by the Leverhulme Trust (Research Project Grant number RPG-2016-194) and by the EPSRC (Standard Grant EP/R015104/1).

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N2 - 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.

AB - 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.

KW - Iterated function system

KW - Self-affline set

KW - Dimension spectrum

KW - Hausdorff dimension

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DO - 10.1007/s00029-021-00674-x

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JO - Selecta Mathematica: New Series

JF - Selecta Mathematica: New Series

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ER -

Dimension spectrum of infinite self-affine iterated function systems

Abstract

Keywords

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Fourier analytic techniques: Fourier analytic techniques in geometry and analysis

Cite this