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- Title
Dimension Conservation for Self-Similar Sets and Fractal Percolation.
- Authors
Falconer, Kenneth; Xiong Jin
- 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 dimH K >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-1 x) > dimH K - 1 - ϵ for a set of x ϵ 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.
- Subjects
FRACTAL analysis; PERCOLATION theory; SELF-similar processes; SET theory; DIMENSIONS; HAUSDORFF measures
- Publication
IMRN: International Mathematics Research Notices, 2015, Vol 2015, Issue 24, p13260
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnv103