The dimensions of inhomogeneous self-affine sets

Stuart Andrew Burrell; Jonathan MacDonald Fraser

doi:10.5186/aasfm.2020.4516

The dimensions of inhomogeneous self-affine sets

Stuart Andrew Burrell, Jonathan MacDonald Fraser

Pure Mathematics

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

Abstract

We prove that the upper box dimension of an inhomogeneous self-affine set is bounded above by the maximum of the affinity dimension and the dimension of the condensation set. In addition, we determine sufficient conditions for this upper bound to be attained, which, in part, constitutes an exploration of the capacity for the condensation set to mitigate dimension drop between the affinity dimension and the corresponding homogeneous attractor. Our work improves and unifies previous results on general inhomogeneous attractors, low-dimensional affine systems, and inhomogeneous self-affine carpets, while providing inhomogeneous analogues of Falconer’s seminal results on homogeneous self-affine sets.

Original language	English
Article number	313-324
Journal	Annales Academiae Scientiarum Fennicae-Mathematica
Volume	45
Issue number	1
Early online date	23 Dec 2019
DOIs	https://doi.org/10.5186/aasfm.2020.4516
Publication status	Published - 2020

Keywords

Inhomogenous attractor
Self-affine set
Box dimension
Affinity dimension

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10.5186/aasfm.2020.4516

vol45pp0313-0324
Copyright © 2020 by Academia Scientiarum Fennica. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the final published version of the work, which was originally published at https://doi.org/10.5186/aasfm.2020.4516
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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
Fractal Geometry and Dimension: Fractal Geometry and dimension theory
Fraser, J. (PI)
The Leverhulme Trust
1/09/16 → 30/06/18
Project: Fellowship

Cite this

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title = "The dimensions of inhomogeneous self-affine sets",

abstract = "We prove that the upper box dimension of an inhomogeneous self-affine set is bounded above by the maximum of the affinity dimension and the dimension of the condensation set. In addition, we determine sufficient conditions for this upper bound to be attained, which, in part, constitutes an exploration of the capacity for the condensation set to mitigate dimension drop between the affinity dimension and the corresponding homogeneous attractor. Our work improves and unifies previous results on general inhomogeneous attractors, low-dimensional affine systems, and inhomogeneous self-affine carpets, while providing inhomogeneous analogues of Falconer{\textquoteright}s seminal results on homogeneous self-affine sets.",

keywords = "Inhomogenous attractor, Self-affine set, Box dimension, Affinity dimension",

author = "Burrell, {Stuart Andrew} and Fraser, {Jonathan MacDonald}",

note = "Funding: SAB thanks the Carnegie Trust for financially supporting this work. JMF was financially supported by a Leverhulme Trust Research Fellowship (RF-2016-500) and an EPSRC Standard Grant (EP/R015104/1).",

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AU - Burrell, Stuart Andrew

AU - Fraser, Jonathan MacDonald

N1 - Funding: SAB thanks the Carnegie Trust for financially supporting this work. JMF was financially supported by a Leverhulme Trust Research Fellowship (RF-2016-500) and an EPSRC Standard Grant (EP/R015104/1).

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N2 - We prove that the upper box dimension of an inhomogeneous self-affine set is bounded above by the maximum of the affinity dimension and the dimension of the condensation set. In addition, we determine sufficient conditions for this upper bound to be attained, which, in part, constitutes an exploration of the capacity for the condensation set to mitigate dimension drop between the affinity dimension and the corresponding homogeneous attractor. Our work improves and unifies previous results on general inhomogeneous attractors, low-dimensional affine systems, and inhomogeneous self-affine carpets, while providing inhomogeneous analogues of Falconer’s seminal results on homogeneous self-affine sets.

AB - We prove that the upper box dimension of an inhomogeneous self-affine set is bounded above by the maximum of the affinity dimension and the dimension of the condensation set. In addition, we determine sufficient conditions for this upper bound to be attained, which, in part, constitutes an exploration of the capacity for the condensation set to mitigate dimension drop between the affinity dimension and the corresponding homogeneous attractor. Our work improves and unifies previous results on general inhomogeneous attractors, low-dimensional affine systems, and inhomogeneous self-affine carpets, while providing inhomogeneous analogues of Falconer’s seminal results on homogeneous self-affine sets.

KW - Inhomogenous attractor

KW - Self-affine set

KW - Box dimension

KW - Affinity dimension

U2 - 10.5186/aasfm.2020.4516

DO - 10.5186/aasfm.2020.4516

M3 - Article

SN - 1239-629X

VL - 45

JO - Annales Academiae Scientiarum Fennicae-Mathematica

JF - Annales Academiae Scientiarum Fennicae-Mathematica

IS - 1

M1 - 313-324

ER -

The dimensions of inhomogeneous self-affine sets

Abstract

Keywords

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Projects

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis

Fractal Geometry and Dimension: Fractal Geometry and dimension theory

Cite this