Dimension (in)equalities and Hölder continuous curves in fractal percolation

Details Dimension (in)equalities and Hölder continuous curves in fractal percolation Dimension (in)equalities and Hölder continuous curves in fractal percolation

2011-11-29

arxiv.org/abs/1111.6848v2

Mathematics

Probability

22 pages, 3 figures, accepted for publication in Journal of Theoretical Probability

Scientific paper

10.1007/s10959-012-0413-8

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 a.s. the union of non-trivial H\"older 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.

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