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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The visible part of plane self-similar sets


Authors: Kenneth J. Falconer and Jonathan M. Fraser
Journal: Proc. Amer. Math. Soc. 141 (2013), 269-278
MSC (2010): Primary 28A78
Published electronically: May 16, 2012
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Abstract: Given a compact subset $ F$ of $ \mathbb{R}^2$, the visible part $ V_\theta F$ of $ F$ from direction $ \theta $ is the set of $ x$ in $ F$ such that the half-line from $ x$ in direction $ \theta $ intersects $ F$ only at $ x$. It is suggested that if $ \dim _H F \geq 1$, then $ \dim _H V_\theta F = 1$ for almost all $ \theta $, where $ \dim _H$ denotes Hausdorff dimension. We confirm this when $ F$ is a self-similar set satisfying the convex open set condition and such that the orthogonal projection of $ F$ onto every line is an interval. In particular the underlying similarities may involve arbitrary rotations and $ F$ need not be connected.


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Additional Information

Kenneth J. Falconer
Affiliation: Mathematical Institute, University of St. Andrews, North Haugh, St. Andrews, Fife KY16 9SS, Scotland
Email: kjf@st-and.ac.uk

Jonathan M. Fraser
Affiliation: Mathematical Institute, University of St. Andrews, North Haugh, St. Andrews, Fife KY16 9SS, Scotland
Email: jmf32@st-and.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11312-7
PII: S 0002-9939(2012)11312-7
Received by editor(s): March 25, 2011
Received by editor(s) in revised form: June 14, 2011
Published electronically: May 16, 2012
Additional Notes: The second author was supported by an EPSRC Doctoral Training Grant
Communicated by: Tatiana Toro
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.