Pointwise Assouad dimension for measures

Roope Anttila

doi:10.1017/prm.2022.83

Pointwise Assouad dimension for measures

Roope Anttila^*

^*Corresponding author for this work

School of Mathematics and Statistics

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

Abstract

We introduce a pointwise variant of the Assouad dimension for measures on metric spaces, and study its properties in relation to the global Assouad dimension. We show that, in general, the value of the pointwise Assouad dimension may differ from the global counterpart, but in many classical cases, the pointwise Assouad dimension exhibits similar exact dimensionality properties as the classical local dimension, namely it equals the global Assouad dimension almost everywhere. We also prove an explicit formula for the Assouad dimension of certain invariant measures with place-dependent probabilities supported on self-conformal sets.

Original language	English
Pages (from-to)	2053-2078
Number of pages	26
Journal	Proceedings of the Royal Society of Edinburgh, Section A: Mathematics
Volume	153
Issue number	6
Early online date	21 Dec 2022
DOIs	https://doi.org/10.1017/prm.2022.83
Publication status	Published - 1 Dec 2023

Keywords

Pointwise doubling measure
Pointwise Assouad dimension
Quasi-Bernoulli measure
Self-conformal set
Place-dependent probabilities

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10.1017/prm.2022.83

https://oulurepo.oulu.fi/handle/10024/46493

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title = "Pointwise Assouad dimension for measures",

abstract = "We introduce a pointwise variant of the Assouad dimension for measures on metric spaces, and study its properties in relation to the global Assouad dimension. We show that, in general, the value of the pointwise Assouad dimension may differ from the global counterpart, but in many classical cases, the pointwise Assouad dimension exhibits similar exact dimensionality properties as the classical local dimension, namely it equals the global Assouad dimension almost everywhere. We also prove an explicit formula for the Assouad dimension of certain invariant measures with place-dependent probabilities supported on self-conformal sets.",

keywords = "Pointwise doubling measure, Pointwise Assouad dimension, Quasi-Bernoulli measure, Self-conformal set, Place-dependent probabilities",

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N2 - We introduce a pointwise variant of the Assouad dimension for measures on metric spaces, and study its properties in relation to the global Assouad dimension. We show that, in general, the value of the pointwise Assouad dimension may differ from the global counterpart, but in many classical cases, the pointwise Assouad dimension exhibits similar exact dimensionality properties as the classical local dimension, namely it equals the global Assouad dimension almost everywhere. We also prove an explicit formula for the Assouad dimension of certain invariant measures with place-dependent probabilities supported on self-conformal sets.

AB - We introduce a pointwise variant of the Assouad dimension for measures on metric spaces, and study its properties in relation to the global Assouad dimension. We show that, in general, the value of the pointwise Assouad dimension may differ from the global counterpart, but in many classical cases, the pointwise Assouad dimension exhibits similar exact dimensionality properties as the classical local dimension, namely it equals the global Assouad dimension almost everywhere. We also prove an explicit formula for the Assouad dimension of certain invariant measures with place-dependent probabilities supported on self-conformal sets.

KW - Pointwise doubling measure

KW - Pointwise Assouad dimension

KW - Quasi-Bernoulli measure

KW - Self-conformal set

KW - Place-dependent probabilities

UR - https://www.scopus.com/pages/publications/85178250754

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JO - Proceedings of the Royal Society of Edinburgh, Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh, Section A: Mathematics

IS - 6

ER -

Pointwise Assouad dimension for measures

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Keywords

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