Dimension conservation for self-similar sets and fractal percolation
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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 line 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.
Falconer , K J & Jin , X 2015 , ' Dimension conservation for self-similar sets and fractal percolation ' International Mathematics Research Notices , vol 2015 , no. 24 , pp. 13260-13289 . DOI: 10.1093/imrn/rnv103
International Mathematics Research Notices
Copyright The Author(s) 2015. Published by Oxford University Press. All rights reserved. This is a pre-copyedited, author-produced PDF of an article accepted for publication in International Mathematics Research Notices following peer review. The version of record Dimension conservation for self-similar sets and fractal percolation Falconer, K. J. & Jin, X. 2015 In : International Mathematics Research Notices. is available online at: http://imrn.oxfordjournals.org/content/early/2015/04/16/imrn.rnv103.abstract
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