The Assouad dimension of randomly generated fractals

Jonathan M. Fraser; Jun Jie Miao; Sascha Troscheit

doi:10.1017/etds.2016.64

The Assouad dimension of randomly generated fractals

Jonathan M. Fraser, Jun Jie Miao, Sascha Troscheit

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

120 Downloads (Pure)

Abstract

We consider several different models for generating random fractals including random self-similar sets, random self-affine carpets, and fractal percolation. In each setting we compute either the \emph{almost sure} or the \emph{Baire typical} Assouad dimension and consider some illustrative examples. Our results reveal a common phenomenon in all 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.

Original language	English
Journal	Ergodic Theory and Dynamical Systems
Early online date	22 Sept 2016
DOIs	https://doi.org/10.1017/etds.2016.64
Publication status	Published - 2016

Keywords

math.MG
math.DS
math.GN
math.PR
28A80, 60J80, 37C45, 54E52, 82B43

Access to Document

10.1017/etds.2016.64

47175604.assouad_randomAccepted author manuscript, 639 KB

Cite this

@article{e2d4b6197c4f4e949856c4f398af5865,

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

abstract = "We consider several different models for generating random fractals including random self-similar sets, random self-affine carpets, and fractal percolation. In each setting we compute either the \emph{almost sure} or the \emph{Baire typical} Assouad dimension and consider some illustrative examples. Our results reveal a common phenomenon in all 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.",

keywords = "math.MG, math.DS, math.GN, math.PR, 28A80, 60J80, 37C45, 54E52, 82B43",

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

note = "26 pages, 7 figures, v3 corrected error in the proof of Theorem 3.2 and sharpened results on exceptional sets",

year = "2016",

doi = "10.1017/etds.2016.64",

language = "English",

journal = "Ergodic Theory and Dynamical Systems",

issn = "0143-3857",

publisher = "Cambridge University Press",

}

TY - JOUR

T1 - The Assouad dimension of randomly generated fractals

AU - Fraser, Jonathan M.

AU - Miao, Jun Jie

AU - Troscheit, Sascha

N1 - 26 pages, 7 figures, v3 corrected error in the proof of Theorem 3.2 and sharpened results on exceptional sets

PY - 2016

Y1 - 2016

N2 - We consider several different models for generating random fractals including random self-similar sets, random self-affine carpets, and fractal percolation. In each setting we compute either the \emph{almost sure} or the \emph{Baire typical} Assouad dimension and consider some illustrative examples. Our results reveal a common phenomenon in all 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.

AB - We consider several different models for generating random fractals including random self-similar sets, random self-affine carpets, and fractal percolation. In each setting we compute either the \emph{almost sure} or the \emph{Baire typical} Assouad dimension and consider some illustrative examples. Our results reveal a common phenomenon in all 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.

KW - math.MG

KW - math.DS

KW - math.GN

KW - math.PR

KW - 28A80, 60J80, 37C45, 54E52, 82B43

U2 - 10.1017/etds.2016.64

DO - 10.1017/etds.2016.64

M3 - Article

SN - 0143-3857

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

ER -

The Assouad dimension of randomly generated fractals

Abstract

Keywords

Access to Document

Fingerprint

Cite this