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Dimension (in)equalities and Hölder 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 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.

preprint2012arXivOpen access
Erik BromanFederico CamiaMatthijs JoostenRonald Meester
math.PR
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Open access4 authors1 topic
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Erik BromanFederico CamiaMatthijs JoostenRonald Meester

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