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arXiv · 1111.6848

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

Erik Broman·Federico Camia·Matthijs Joosten·Ronald Meester
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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.

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Erik Broman, Federico Camia, Matthijs Joosten, Ronald Meester. 2012-03-07. Dimension (in)equalities and Hölder continuous curves in fractal percolation. https://doi.org/10.1007/s10959-012-0413-8

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