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Assouad Spectrum Thresholds for Some Random Constructions

Part of: Smooth dynamical systems: general theory Markov processes Classical measure theory

Published online by Cambridge University Press:  12 December 2019

Sascha Troscheit
Sascha Troscheit*
Affiliation:
Faculty of Mathematics, University of Vienna, Oskar Morgenstern Platz 1, 1090Wien, Austria Email: sascha.troscheit@univie.ac.at URL: https://www.mat.univie.ac.at/∼troscheit/
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Abstract

The Assouad dimension of a metric space determines its extremal scaling properties. The derived notion of the Assouad spectrum fixes relative scales by a scaling function to obtain interpolation behaviour between the quasi-Assouad and the box-counting dimensions. While the quasi-Assouad and Assouad dimensions often coincide, they generally differ in random constructions. In this paper we consider a generalised Assouad spectrum that interpolates between the quasi-Assouad and the Assouad dimension. For common models of random fractal sets, we obtain a dichotomy of its behaviour by finding a threshold function where the quasi-Assouad behaviour transitions to the Assouad dimension. This threshold can be considered a phase transition, and we compute the threshold for the Gromov boundary of Galton–Watson trees and one-variable random self-similar and self-affine constructions. We describe how the stochastically self-similar model can be derived from the Galton–Watson tree result.

Keywords

Assouad dimensionlocal complexityGalton–Watson processstochastic self-similarity

MSC classification

Primary: 28A80: Fractals
Secondary: 37C45: Dimension theory of dynamical systems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 2 , June 2020 , pp. 434 - 453
Copyright
© Canadian Mathematical Society 2019

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Footnotes

The author was initially supported by NSERC Grants 2014-03154 and 2016-03719, and the University of Waterloo.

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