Dimension (in)equalities and Hölder continuous curves in fractal percolation
arXiv:1111.6848 · doi:10.1007/s10959-012-0413-8
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 a.s. 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.
22 pages, 3 figures, accepted for publication in Journal of Theoretical Probability