Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-25T10:25:31.779Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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
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

References

1.Alon, N., Spencer, J. H. and Erdőos, P., The probabilistic method (Wiley New York, 1992).Google Scholar
2.Chayes, L., Aspects of the fractal percolation process, in Fractal geometry and stochastics (ed. Bandt, C., Graf, S. and Zähle, M.), Progress in Probability (Birkhäuser, Basel, 1995).Google Scholar
3.Chayes, J. T., Chayes, L. and Durrett, R., Connectivity properties of Mandelbrot's percolation process, Prob. Theory Relat. Fields 77 (1988), 307–324.CrossRefGoogle Scholar
4.Csörnyei, M., On the visibility of invisible sets, Annales Acad. Sci. Fenn. Math. 25 (2000), 417–421.Google Scholar
5.Falconer, K. J. and Fraser, J. M., The visible part of plane self-similar sets, Proc. Am. Math. Soc., in press (eprint arXiv:1004.5067v1 [math.MG]).Google Scholar
6.Falconer, K. J. and Grimmett, G. R., On the geometry of random Cantor sets and fractal percolation, J. Theor. Prob. 5 (1992), 465–485 (Correction, J. Theor. Prob. 7 (1994), 209–210).CrossRefGoogle Scholar
7.Grimmett, G., Percolation, Second Edition (Springer, 1999).CrossRefGoogle Scholar
8.Järvenpää, M., Visibility and fractal percolation, Rev. Roumaine Math. Pures Appl. 54 (2009), 451–459.Google Scholar
9.Järvenpää, E., Järvenpää, M., MacManus, P. and O'Neil, T. C., Visible parts and dimensions, Nonlinearity 16 (2003), 803–818.CrossRefGoogle Scholar
10.Mattila, P., Geometry of sets and measures in Euclidean spaces: fractals and rectifiability (Cambridge University Press, 1995).CrossRefGoogle Scholar
11.Mattila, P., Hausdorff dimension, projections, and the Fourier transform, Publ. Mat. 48 (2004), 3–48.CrossRefGoogle Scholar
12.O'Neil, T. C., The Hausdorff dimension of visible sets of planar continua, Trans. Am. Math. Soc. 359 (2007), 5141–5170.CrossRefGoogle Scholar
13.Peres, Y. and Schlag, W., Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions, Duke Math. J. 102 (2000), 193–251.CrossRefGoogle Scholar
14.Rams, M. and Simon, K., Projections of fractal percolations, preprint.Google Scholar