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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 [email protected]
JUN JIE MIAO
Affiliation:
Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, PR China email [email protected]
SASCHA TROSCHEIT
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, UK email [email protected]
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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.

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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