Dimension conservation for self-similar sets and fractal percolation

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Abstract

We introduce a technique that uses projection properties of fractal percolation to establish dimension conservation results for sections of deterministic self-similar sets. For example, let K be a self-similar subset of R2 with Hausdorff dimension dimHK >1 such that the rotational components of the underlying similarities generate the full rotation group. Then, for all ε >0, writing πθ for projection onto the Lθ in direction θ, the Hausdorff dimensions of the sections satisfy dimH (K ∩ πθ-1x)> dimHK - 1 - ε for a set of x ∈ Lθ of positive Lebesgue measure, for all directions θ except for those in a set of Hausdorff dimension 0. For a class of self-similar sets we obtain a similar conclusion for all directions, but with lower box dimension replacing Hausdorff dimensions of sections. We obtain similar inequalities for the dimensions of sections of Mandelbrot percolation sets.

Original languageEnglish
Pages (from-to)13260-13289
Number of pages30
JournalInternational Mathematics Research Notices
Volume2015
Issue number24
Early online date15 Apr 2015
DOIs
Publication statusPublished - 2015

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