Dimension conservation for self-similar sets and fractal percolation
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.
Citation
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 . https://doi.org/10.1093/imrn/rnv103
Publication
International Mathematics Research Notices
Status
Peer reviewed
ISSN
1073-7928Type
Journal article
Collections
Items in the St Andrews Research Repository are protected by copyright, with all rights reserved, unless otherwise indicated.