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Pointwise Assouad dimension for measures

Part of: Set functions and measures on spaces with additional structure Classical measure theory

Published online by Cambridge University Press:  21 December 2022

Roope Anttila Open the ORCID record for Roope Anttila [Opens in a new window]
Roope Anttila*
Affiliation:
Research Unit of Mathematical Sciences, University of Oulu, P.O.Box 8000, Oulu, FI-90014, Finland roope.anttila@oulu.fi
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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 measurepointwise Assouad dimensionquasi-Bernoulli measureself-conformal setplace-dependent probabilities

MSC classification

Primary: 28A80: Fractals
Secondary: 28C15: Set functions and measures on topological spaces (regularity of measures, etc.)

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 6 , December 2023 , pp. 2053 - 2078
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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