1 research outputs found
Dimension (in)equalities and H\"older continuous curves in fractal percolation
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.Comment: 22 pages, 3 figures, accepted for publication in Journal of
Theoretical Probabilit