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Geometry and Analysis of Fractals pp 303–323Cite as

The Geometry of Fractal Percolation

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 88))

Abstract

A well studied family of random fractals called fractal percolation is discussed. We focus on the projections of fractal percolation on the plane. Our goal is to present stronger versions of the classical Marstrand theorem, valid for almost every realization of fractal percolation. The extensions go in three directions:

\(\bullet \) the statements work for all directions, not almost all,

\(\bullet \) the statements are true for more general projections, for example radial projections onto a circle,

\(\bullet \) in the case \(\dim _H >1\), each projection has not only positive Lebesgue measure but also has nonempty interior.

Rams was partially supported by the MNiSW grant N201 607640 (Poland). The research of Simon was supported by OTKA Foundation # K 104745.

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Authors and Affiliations

  1. Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, Warsaw, 00-956, Poland

    Michał Rams

  2. Institute of Mathematics, Technical University of Budapest, Budapest, H-1529, B.O.box 91, Hungary

    Károly Simon

Authors
  1. Michał Rams
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  2. Károly Simon
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Correspondence to Károly Simon .

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Editors and Affiliations

  1. Department of Mathematics, The Chinese University of Hong Kong, Hongkong, Hong Kong SAR

    De-Jun Feng

  2. Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, Hong Kong SAR

    Ka-Sing Lau

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Rams, M., Simon, K. (2014). The Geometry of Fractal Percolation. In: Feng, DJ., Lau, KS. (eds) Geometry and Analysis of Fractals. Springer Proceedings in Mathematics & Statistics, vol 88. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43920-3_11

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