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

Erik I. Broman (), Federico Camia (), Matthijs Joosten () and Ronald Meester ()
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Erik I. Broman: Uppsala Universitet
Federico Camia: VU University Amsterdam
Matthijs Joosten: VU University Amsterdam
Ronald Meester: VU University Amsterdam

Journal of Theoretical Probability, 2013, vol. 26, issue 3, 836-854

Abstract: 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.

Keywords: Fractal percolation; Hausdorff dimension; Box counting dimension; Hölder continuous curves; Subsequential weak limits; 60K35; 28A80; 37F35; 54C05 (search for similar items in EconPapers)
Date: 2013
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10959-012-0413-8

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