The Assouad dimension of randomly generated fractals

Jonathan MacDonald Fraser; Jun Jie Miao; Sascha Troscheit

doi:10.1017/etds.2016.64

The Assouad dimension of randomly generated fractals

Jonathan MacDonald Fraser, Jun Jie Miao, Sascha Troscheit

Pure Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We consider several dierent 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 Hausdor or packing dimension.

Original language	English
Pages (from-to)	982-1011
Journal	Ergodic Theory and Dynamical Systems
Volume	38
Issue number	3
Early online date	22 Sept 2016
DOIs	https://doi.org/10.1017/etds.2016.64
Publication status	Published - May 2018

Keywords

Assouad dimension
Random fractal
Self-similar set
Self-affine carpet
Mandelbrot percolation
Baire category

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10.1017/etds.2016.64

Fraser_2016_Assouad_ETDS_AAM
© Cambridge University Press, 2016. This work is made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at www.cambridge.org / http://dx.doi.org/10.1017/etds.2016.64
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Cite this

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title = "The Assouad dimension of randomly generated fractals",

abstract = "We consider several dierent 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 Hausdor or packing dimension.",

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author = "Fraser, {Jonathan MacDonald} and Miao, {Jun Jie} and Sascha Troscheit",

note = "JMF was fi nancially supported by the EPSRC grant EP/J013560/1 whilst employed at the University of Warwick. JJM was partially supported by the NNSF of China (no. 11201152), the Fund for the Doctoral Program of Higher Education of China (no. 20120076120001) and SRF for ROCS, SEM (no. 01207427) ST was financially supported by the EPSRC Doctoral Training Grant EP/K503162/1.",

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The Assouad dimension of randomly generated fractals

Abstract

Keywords

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