The Assouad dimension of self-affine measures on sponges

Jonathan Fraser; Istvan Tamas Kolossvary

doi:10.1017/etds.2022.64

The Assouad dimension of self-affine measures on sponges

Jonathan Fraser, Istvan Tamas Kolossvary^*

^*Corresponding author for this work

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

Abstract

We derive upper and lower bounds for the Assouad and lower dimensions of self-affine measures in ℝ^d generated by diagonal matrices and satisfying suitable separation conditions. The upper and lower bounds always coincide for d=2,3 , yielding precise explicit formulae for those dimensions. Moreover, there are easy-to-check conditions guaranteeing that the bounds coincide for d⩾4 . An interesting consequence of our results is that there can be a ‘dimension gap’ for such self-affine constructions, even in the plane. That is, we show that for some self-affine carpets of ‘Barański type’ the Assouad dimension of all associated self-affine measures strictly exceeds the Assouad dimension of the carpet by some fixed δ>0 depending only on the carpet. We also provide examples of self-affine carpets of ‘Barański type’ where there is no dimension gap and in fact the Assouad dimension of the carpet is equal to the Assouad dimension of a carefully chosen self-affine measure.

Original language	English
Number of pages	23
Journal	Ergodic Theory and Dynamical Systems
Volume	FirstView
Early online date	8 Sept 2022
DOIs	https://doi.org/10.1017/etds.2022.64
Publication status	E-pub ahead of print - 8 Sept 2022

Keywords

Assouad dimension
Lower dimension
Self-affine carpet
Self-affine sponge
Dimension gap

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10.1017/etds.2022.64Licence: CC BY

Fraser_2022_ETDS_Assouad-Dimension_CC
Copyright © The Author(s), 2022. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Final published version, 460 KBLicence: CC BY

2 Finished

New perspectives in the dimension: New perspectives in the dimension theory of fractals
Fraser, J. (PI)
The Leverhulme Trust
1/09/19 → 31/01/23
Project: Standard
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

1 Preprint

The Assouad dimension of self-affine measures on sponges
Fraser, J. M. & Kolossváry, I., 21 Mar 2022, (Submitted).
Research output: Working paper › Preprint

Cite this

@article{c2919de9f96c462ea4a9876a616fc8f1,

title = "The Assouad dimension of self-affine measures on sponges",

abstract = "We derive upper and lower bounds for the Assouad and lower dimensions of self-affine measures in ℝd generated by diagonal matrices and satisfying suitable separation conditions. The upper and lower bounds always coincide for d=2,3 , yielding precise explicit formulae for those dimensions. Moreover, there are easy-to-check conditions guaranteeing that the bounds coincide for d⩾4 . An interesting consequence of our results is that there can be a {\textquoteleft}dimension gap{\textquoteright} for such self-affine constructions, even in the plane. That is, we show that for some self-affine carpets of {\textquoteleft}Bara{\'n}ski type{\textquoteright} the Assouad dimension of all associated self-affine measures strictly exceeds the Assouad dimension of the carpet by some fixed δ>0 depending only on the carpet. We also provide examples of self-affine carpets of {\textquoteleft}Bara{\'n}ski type{\textquoteright} where there is no dimension gap and in fact the Assouad dimension of the carpet is equal to the Assouad dimension of a carefully chosen self-affine measure.",

keywords = "Assouad dimension, Lower dimension, Self-affine carpet, Self-affine sponge, Dimension gap",

author = "Jonathan Fraser and Kolossvary, {Istvan Tamas}",

note = "Funding: Royal Society of Edinburgh - 70249; Leverhulme Trust - RPG-2019-034; Engineering and Physical Sciences Research Council - EP/R015104/1.",

year = "2022",

month = sep,

day = "8",

doi = "10.1017/etds.2022.64",

language = "English",

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T1 - The Assouad dimension of self-affine measures on sponges

AU - Fraser, Jonathan

AU - Kolossvary, Istvan Tamas

N1 - Funding: Royal Society of Edinburgh - 70249; Leverhulme Trust - RPG-2019-034; Engineering and Physical Sciences Research Council - EP/R015104/1.

PY - 2022/9/8

Y1 - 2022/9/8

N2 - We derive upper and lower bounds for the Assouad and lower dimensions of self-affine measures in ℝd generated by diagonal matrices and satisfying suitable separation conditions. The upper and lower bounds always coincide for d=2,3 , yielding precise explicit formulae for those dimensions. Moreover, there are easy-to-check conditions guaranteeing that the bounds coincide for d⩾4 . An interesting consequence of our results is that there can be a ‘dimension gap’ for such self-affine constructions, even in the plane. That is, we show that for some self-affine carpets of ‘Barański type’ the Assouad dimension of all associated self-affine measures strictly exceeds the Assouad dimension of the carpet by some fixed δ>0 depending only on the carpet. We also provide examples of self-affine carpets of ‘Barański type’ where there is no dimension gap and in fact the Assouad dimension of the carpet is equal to the Assouad dimension of a carefully chosen self-affine measure.

AB - We derive upper and lower bounds for the Assouad and lower dimensions of self-affine measures in ℝd generated by diagonal matrices and satisfying suitable separation conditions. The upper and lower bounds always coincide for d=2,3 , yielding precise explicit formulae for those dimensions. Moreover, there are easy-to-check conditions guaranteeing that the bounds coincide for d⩾4 . An interesting consequence of our results is that there can be a ‘dimension gap’ for such self-affine constructions, even in the plane. That is, we show that for some self-affine carpets of ‘Barański type’ the Assouad dimension of all associated self-affine measures strictly exceeds the Assouad dimension of the carpet by some fixed δ>0 depending only on the carpet. We also provide examples of self-affine carpets of ‘Barański type’ where there is no dimension gap and in fact the Assouad dimension of the carpet is equal to the Assouad dimension of a carefully chosen self-affine measure.

KW - Assouad dimension

KW - Lower dimension

KW - Self-affine carpet

KW - Self-affine sponge

KW - Dimension gap

U2 - 10.1017/etds.2022.64

DO - 10.1017/etds.2022.64

M3 - Article

SN - 0143-3857

VL - FirstView

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

ER -

The Assouad dimension of self-affine measures on sponges

Abstract

Keywords

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Projects

New perspectives in the dimension: New perspectives in the dimension theory of fractals

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis

Research output

The Assouad dimension of self-affine measures on sponges

Cite this