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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Assouad type dimensions and homogeneity of fractals


Author: Jonathan M. Fraser
Journal: Trans. Amer. Math. Soc. 366 (2014), 6687-6733
MSC (2010): Primary 28A80; Secondary 28A78, 28A20, 28C15
Published electronically: May 13, 2014
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Abstract: We investigate several aspects of the Assouad dimension and the lower dimension, which together form a natural `dimension pair'. In particular, we compute these dimensions for certain classes of self-affine sets and quasi-self-similar sets and study their relationships with other notions of dimension, such as the Hausdorff dimension for example. We also investigate some basic properties of these dimensions including their behaviour regarding unions and products and their set theoretic complexity.


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Additional Information

Jonathan M. Fraser
Affiliation: Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, Scotland
Address at time of publication: Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL, United Kingdom
Email: jmf32@st-andrews.ac.uk, jon.fraser32@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9947-2014-06202-8
PII: S 0002-9947(2014)06202-8
Keywords: Assouad dimension, lower dimension, self-affine carpet, Ahlfors regular, measurability, Baire hierarchy
Received by editor(s): January 18, 2013
Received by editor(s) in revised form: May 5, 2013
Published electronically: May 13, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.