On the upper regularity dimensions of measures

Jonathan MacDonald Fraser; Douglas Charles Howroyd

doi:10.1512/iumj.2020.69.7840

On the upper regularity dimensions of measures

Jonathan MacDonald Fraser, Douglas Charles Howroyd

Pure Mathematics

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

Abstract

We study the upper regularity dimension which describes the extremal local scaling behaviour of a measure and effectively quantifies the notion of doubling. We conduct a thorough study of the upper regularity dimension, including its relationship with other concepts such as the Assouad dimension, the upper local dimension, the L^q-spectrum and weak tangent measures. We also compute the upper regularity dimension explicitly in a number of important contexts including self-similar measures, self-affine measures, and measures on sequences.

Original language	English
Pages (from-to)	685–712
Number of pages	22
Journal	Indiana University Mathematics Journal
Volume	69
Issue number	2
DOIs	https://doi.org/10.1512/iumj.2020.69.7840
Publication status	Published - 1 Feb 2020

Keywords

Upper regularity dimension
Assouad dimension
Local dimension
Self-similar measure
Self-affine measure
Doubling measure
Weak tangent
Lq-spectrum

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10.1512/iumj.2020.69.7840

https://arxiv.org/abs/1706.09340Licence: Other

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title = "On the upper regularity dimensions of measures",

abstract = "We study the upper regularity dimension which describes the extremal local scaling behaviour of a measure and effectively quantifies the notion of doubling. We conduct a thorough study of the upper regularity dimension, including its relationship with other concepts such as the Assouad dimension, the upper local dimension, the Lq-spectrum and weak tangent measures. We also compute the upper regularity dimension explicitly in a number of important contexts including self-similar measures, self-affine measures, and measures on sequences.",

keywords = "Upper regularity dimension, Assouad dimension, Local dimension, Self-similar measure, Self-affine measure, Doubling measure, Weak tangent, Lq-spectrum",

author = "Fraser, \{Jonathan MacDonald\} and Howroyd, \{Douglas Charles\}",

note = "Funding: J.M.F. was financially supported by a Leverhulme Trust Research Fellowship (RF-2016-500) and D.C.H. was financially supported by an EPSRC Doctoral Training Grant (EP/N509759/1).",

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AU - Howroyd, Douglas Charles

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N2 - We study the upper regularity dimension which describes the extremal local scaling behaviour of a measure and effectively quantifies the notion of doubling. We conduct a thorough study of the upper regularity dimension, including its relationship with other concepts such as the Assouad dimension, the upper local dimension, the Lq-spectrum and weak tangent measures. We also compute the upper regularity dimension explicitly in a number of important contexts including self-similar measures, self-affine measures, and measures on sequences.

AB - We study the upper regularity dimension which describes the extremal local scaling behaviour of a measure and effectively quantifies the notion of doubling. We conduct a thorough study of the upper regularity dimension, including its relationship with other concepts such as the Assouad dimension, the upper local dimension, the Lq-spectrum and weak tangent measures. We also compute the upper regularity dimension explicitly in a number of important contexts including self-similar measures, self-affine measures, and measures on sequences.

KW - Upper regularity dimension

KW - Assouad dimension

KW - Local dimension

KW - Self-similar measure

KW - Self-affine measure

KW - Doubling measure

KW - Weak tangent

KW - Lq-spectrum

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VL - 69

SP - 685

EP - 712

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

IS - 2

ER -

On the upper regularity dimensions of measures

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Fractal Geometry and Dimension: Fractal Geometry and dimension theory

On the regularity dimensions of measures

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On the upper regularity dimensions of measures

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Keywords

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Fractal Geometry and Dimension: Fractal Geometry and dimension theory

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On the regularity dimensions of measures

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