Dimension (in)equalities and H\"older continuous curves in fractal percolation
Probability
2012-03-08 v2
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 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.
Keywords
Cite
@article{arxiv.1111.6848,
title = {Dimension (in)equalities and H\"older continuous curves in fractal percolation},
author = {Erik Broman and Federico Camia and Matthijs Joosten and Ronald Meester},
journal= {arXiv preprint arXiv:1111.6848},
year = {2012}
}
Comments
22 pages, 3 figures, accepted for publication in Journal of Theoretical Probability