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Overlapping self-affine sets of Kakeya type

Published online by Cambridge University Press:  01 June 2009

ANTTI KÄENMÄKI  and
PABLO SHMERKIN
ANTTI KÄENMÄKI
Affiliation:
Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35 (MaD), FI-40014, Finland (email: antakae@maths.jyu.fi, shmerkin@maths.jyu.fi)
PABLO SHMERKIN
Affiliation:
Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35 (MaD), FI-40014, Finland (email: antakae@maths.jyu.fi, shmerkin@maths.jyu.fi)
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Abstract

We compute the Minkowski dimension for a family of self-affine sets on ℝ2. Our result holds for every (rather than generic) set in the class. Moreover, we exhibit explicit open subsets of this class where we allow overlapping, and do not impose any conditions on the norms of the linear maps. The family under consideration was inspired by the theory of Kakeya sets.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 3 , June 2009 , pp. 941 - 965
Copyright
Copyright © Cambridge University Press 2008

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