The visible part of plane self-similar sets

Kenneth John Falconer; Jonathan Macdonald Fraser

doi:10.1090/S0002-9939-2012-11312-7

The visible part of plane self-similar sets

Kenneth John Falconer, Jonathan Macdonald Fraser

Pure Mathematics

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

Abstract

Given a compact subset F of R², the visible part V_θF of F from direction θ is
the set of x in F such that the half-line from x in direction θ intersects F only at
x. It is suggested that if dim_H F ≥ 1 then dim_H V_θF = 1 for almost all θ , where
dim_H denotes Hausdorff dimension. We conrm 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.

Original language	English
Pages (from-to)	269-278
Journal	Proceedings of the American Mathematical Society
Volume	141
Issue number	1
Early online date	16 May 2012
DOIs	https://doi.org/10.1090/S0002-9939-2012-11312-7
Publication status	Published - 2013

Keywords

Metric Geometry

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10.1090/S0002-9939-2012-11312-7

FalconerProcAmMathSoc2012VisiblePart
© Copyright 2012 American Mathematical Society. First published in Proceedings of the American Mathematical Society 2012, published by the American Mathematical Society)
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http://arxiv.org/pdf/1004.5067.pdf

Cite this

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The visible part of plane self-similar sets

Abstract

Keywords

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