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On the dimension of planar self-affine sets with non-invertible maps

Part of: Classical measure theory

Published online by Cambridge University Press:  07 September 2023

Balázs Bárány Open the ORCID record for Balázs Bárány [Opens in a new window]  and
Viktor Körtvélyesi Open the ORCID record for Viktor Körtvélyesi [Opens in a new window]
Balázs Bárány
Affiliation:
Department of Stochastics, Institute of Mathematics, Budapest University of Technology and Economics, Muűegyetem rkp. 3, H-1111 Budapest, Hungary (balubsheep@gmail.com, vkortvelyesi@gmail.com)
Viktor Körtvélyesi
Affiliation:
Department of Stochastics, Institute of Mathematics, Budapest University of Technology and Economics, Muűegyetem rkp. 3, H-1111 Budapest, Hungary (balubsheep@gmail.com, vkortvelyesi@gmail.com)
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Abstract

In this paper, we study the dimension of planar self-affine sets, of which generating iterated function system (IFS) contains non-invertible affine mappings. We show that under a certain separation condition the dimension equals to the affinity dimension for a typical choice of the linear-parts of the non-invertible mappings, furthermore, we show that the dimension is strictly smaller than the affinity dimension for certain choices of parameters.

Keywords

Self-affine setHausdorff dimensioniterated function system

MSC classification

Primary: 28A80: Fractals
Secondary: 28A78: Hausdorff and packing measures
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 16
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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