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

Erik I. Broman, Federico Camia, Matthijs Joosten, Ronald Meester

Research output: Contribution to journalArticle

Abstract

We relate various concepts of fractal dimension of the limiting set C in fractal percolation to the dimensions of the set consisting of connected components larger than one point and its complement in 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.

Original languageEnglish (US)
Pages (from-to)836-854
Number of pages19
JournalJournal of Theoretical Probability
Volume26
Issue number3
DOIs
StatePublished - Sep 2013

Keywords

  • Box counting dimension
  • Fractal percolation
  • Hausdorff dimension
  • Hölder continuous curves
  • Subsequential weak limits

ASJC Scopus subject areas

  • Statistics and Probability
  • Mathematics(all)
  • Statistics, Probability and Uncertainty

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