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Visible parts of fractal percolation

Published online by Cambridge University Press:  12 April 2012

Ida Arhosalo
Affiliation:
Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35, 40014 University of Jyväskylä, Finland ([email protected])
Esa Järvenpää
Affiliation:
Department of Mathematical Sciences, Department of Mathematical Sciences, PO Box 3000, University of Oulu, Finland ([email protected]; [email protected])
Maarit Järvenpää
Affiliation:
Department of Mathematical Sciences, Department of Mathematical Sciences, PO Box 3000, University of Oulu, Finland ([email protected]; [email protected])
Michal Rams
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, ul. Sniadeckich 8, PO Box 21, 00-956 Warsaw, Poland ([email protected])
Pablo Shmerkin
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK ([email protected])
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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.

MSC classification

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2012

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