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Dimension (In)equalities and Hölder Continuous Curves in Fractal Percolation

Author

Listed:
  • Erik I. Broman

    (Uppsala Universitet)

  • Federico Camia

    (VU University Amsterdam)

  • Matthijs Joosten

    (VU University Amsterdam)

  • Ronald Meester

    (VU University Amsterdam)

Abstract

We relate various concepts of fractal dimension of the limiting set $\mathcal{C}$ in fractal percolation to the dimensions of the set consisting of connected components larger than one point and its complement in $\mathcal{C}$ (the “dust”). In two dimensions, we also show that the set consisting of connected components larger than one point is almost surely the union of non-trivial Hölder continuous curves, all with the same exponent. Finally, we give a short proof of the fact that in two dimensions, any curve in the limiting set must have Hausdorff dimension strictly larger than 1.

Suggested Citation

  • Erik I. Broman & Federico Camia & Matthijs Joosten & Ronald Meester, 2013. "Dimension (In)equalities and Hölder Continuous Curves in Fractal Percolation," Journal of Theoretical Probability, Springer, vol. 26(3), pages 836-854, September.
    • Handle: RePEc:spr:jotpro:v:26:y:2013:i:3:d:10.1007_s10959-012-0413-8
    DOI: 10.1007/s10959-012-0413-8
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    Cited by:

    1. Changhao Chen & Tuomo Ojala & Eino Rossi & Ville Suomala, 2017. "Fractal Percolation, Porosity, and Dimension," Journal of Theoretical Probability, Springer, vol. 30(4), pages 1471-1498, December.

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