Exact dimensionality and projections of random self-similar measures and sets
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We study the geometric properties of random multiplicative cascade measures defined on self-similar sets. We show that such measures and their projections and sections are almost surely exact dimensional, generalizing a result of Feng and Hu's for self-similar measures. This, together with a compact group extension argument, enables us to generalize Hochman and Shmerkin's theorems on projections of deterministic self-similar measures to these random measures without requiring any separation conditions on the underlying sets. We give applications to self-similar sets and fractal percolation, including new results on projections, C1-images and distance sets.
Falconer , K & Jin , X 2014 , ' Exact dimensionality and projections of random self-similar measures and sets ' , Journal of the London Mathematical Society , vol. 90 , no. 2 , pp. 388-412 . https://doi.org/10.1112/jlms/jdu031
Journal of the London Mathematical Society
© 2014. London Mathematical Society. This is the accepted version before publication of the following article: Exact dimensionality and projections of random self-similar measures and sets Falconer, K. & Jin, X. 2014 In : Journal of the London Mathematical Society. 25 p., which has been published in final form at http://jlms.oxfordjournals.org/content/90/2/388