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The Assouad dimension of randomly generated fractals

Published online by Cambridge University Press:  22 September 2016

JONATHAN M. FRASER ,
JUN JIE MIAO  and
SASCHA TROSCHEIT Open the ORCID record for SASCHA TROSCHEIT [Opens in a new window]
JONATHAN M. FRASER
Affiliation:
School of Mathematics, The University of Manchester, Manchester, M13 9PL, UK email jon.fraser32@gmail.com
JUN JIE MIAO
Affiliation:
Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, PR China email jjmiao@math.ecnu.edu.cn
SASCHA TROSCHEIT
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, UK email s.troscheit@st-andrews.ac.uk
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Abstract

We consider several different models for generating random fractals including random self-similar sets, random self-affine carpets, and Mandelbrot percolation. In each setting we compute either the almost sure or the Baire typical Assouad dimension and consider some illustrative examples. Our results reveal a phenomenon common to each of our models: the Assouad dimension of a randomly generated fractal is generically as big as possible and does not depend on the measure-theoretic or topological structure of the sample space. This is in stark contrast to the other commonly studied notions of dimension like the Hausdorff or packing dimension.

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Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 3 , May 2018 , pp. 982 - 1011
Copyright
© Cambridge University Press, 2016 

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