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Lower Assouad Dimension of Measures and Regularity

Part of: Classical measure theory Set functions and measures on spaces with additional structure Smooth dynamical systems: general theory

Published online by Cambridge University Press:  22 November 2019

KATHRYN E. HARE  and
SASCHA TROSCHEIT
KATHRYN E. HARE
Affiliation:
Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Canada, N2L 3G1. e-mail: kehare@uwaterloo.ca
SASCHA TROSCHEIT
Affiliation:
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria. e-mail: sascha.troscheit@univie.ac.at
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Abstract

In analogy with the lower Assouad dimensions of a set, we study the lower Assouad dimensions of a measure. As with the upper Assouad dimensions, the lower Assouad dimensions of a measure provide information about the extreme local behaviour of the measure. We study the connection with other dimensions and with regularity properties. In particular, the quasi-lower Assouad dimension is dominated by the infimum of the measure’s lower local dimensions. Although strict inequality is possible in general, equality holds for the class of self-similar measures of finite type. This class includes all self-similar, equicontractive measures satisfying the open set condition, as well as certain “overlapping” self-similar measures, such as Bernoulli convolutions with contraction factors that are inverses of Pisot numbers.

We give lower bounds for the lower Assouad dimension for measures arising from a Moran construction, prove that self-affine measures are uniformly perfect and have positive lower Assouad dimension, prove that the Assouad spectrum of a measure converges to its quasi-Assouad dimension and show that coincidence of the upper and lower Assouad dimension of a measure does not imply that the measure is s-regular.

MSC classification

Primary: 28C15: Set functions and measures on topological spaces (regularity of measures, etc.)
Secondary: 28A80: Fractals 37C45: Dimension theory of dynamical systems
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 2 , March 2021 , pp. 379 - 415
Copyright
© Cambridge Philosophical Society 2019

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Footnotes

†

Supported by NSERC 2016-03719.

‡

Supported in part by NSERC 2016-03719, NSERC 2014-03154 and the University of Waterloo.

References

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