The visible part of plane self-similar sets
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- by Kenneth J. Falconer and Jonathan M. Fraser PDF
- Proc. Amer. Math. Soc. 141 (2013), 269-278 Request permission
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.References
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Additional Information
- Kenneth J. Falconer
- Affiliation: Mathematical Institute, University of St. Andrews, North Haugh, St. Andrews, Fife KY16 9SS, Scotland
- MR Author ID: 65025
- Email: kjf@st-and.ac.uk
- Jonathan M. Fraser
- Affiliation: Mathematical Institute, University of St. Andrews, North Haugh, St. Andrews, Fife KY16 9SS, Scotland
- MR Author ID: 946983
- Email: jmf32@st-and.ac.uk
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 269-278
- MSC (2010): Primary 28A78
- DOI: https://doi.org/10.1090/S0002-9939-2012-11312-7
- MathSciNet review: 2988729