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Projections of fractal percolations

Published online by Cambridge University Press:  11 September 2013

MICHAŁ RAMS  and
KÁROLY SIMON
MICHAŁ RAMS
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warsaw, Poland email rams@impan.gov.pl
KÁROLY SIMON
Affiliation:
Institute of Mathematics, Technical University of Budapest, H-1529 B.O. Box 91, Hungary email simonk@math.bme.hu
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Abstract

In this paper we study the radial and orthogonal projections and the distance sets of the random Cantor sets $E\subset { \mathbb{R} }^{2} $, which are called Mandelbrot percolation or percolation fractals. We prove that the following assertion holds almost surely: if the Hausdorff dimension of $E$ is greater than $1$ then the orthogonal projection to every line, the radial projection with every centre, and the distance set from every point contain intervals.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 2 , April 2015 , pp. 530 - 545
Copyright
© Cambridge University Press, 2013 

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