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Visible parts of fractal percolation

Part of: Classical measure theory

Published online by Cambridge University Press:  12 April 2012

Ida Arhosalo ,
Esa Järvenpää ,
Maarit Järvenpää ,
Michal Rams  and
Pablo Shmerkin
Ida Arhosalo
Affiliation:
Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35, 40014 University of Jyväskylä, Finland (idmaarho@jyu.fi)
Esa Järvenpää
Affiliation:
Department of Mathematical Sciences, Department of Mathematical Sciences, PO Box 3000, University of Oulu, Finland (esa.jarvenpaa@oulu.fi; maarit.jarvenpaa@oulu.fi)
Maarit Järvenpää
Affiliation:
Department of Mathematical Sciences, Department of Mathematical Sciences, PO Box 3000, University of Oulu, Finland (esa.jarvenpaa@oulu.fi; maarit.jarvenpaa@oulu.fi)
Michal Rams
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, ul. Sniadeckich 8, PO Box 21, 00-956 Warsaw, Poland (rams@impan.pl)
Pablo Shmerkin
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK (pablo.shmerkin@manchester.ac.uk)
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Abstract

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We study dimensional properties of visible parts of fractal percolation in the plane. Provided that the dimension of the fractal percolation is at least 1, we show that, conditioned on non-extinction, almost surely all visible parts from lines are one dimensional. Furthermore, almost all of them have positive and finite Hausdorff measure. We also verify analogous results for visible parts from points. These results are motivated by an open problem on the dimensions of visible parts.

Keywords

visible partfractal percolationHausdorff dimension

MSC classification

Secondary: 28A80: Fractals

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 2 , June 2012 , pp. 311 - 331
Copyright
Copyright © Edinburgh Mathematical Society 2012

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