Local entropy averages and projections of fractal measures
Abstract
We show that for families of measures on Euclidean space which satisfy an ergodictheoretic form of "selfsimilarity" under the operation of rescaling, the dimension of linear images of the measure behaves in a semicontinuous way. We apply this to prove the following conjecture of Furstenberg: Let m,n be integers which are not powers of the same integer, and let X,Y be closed subsets of the unit interval which are invariant, respectively, under timesm mod 1 and timesn mod 1. Then, for any nonzero t: dim(X+tY)=min{1,dim(X)+dim(Y)}. A similar result holds for invariant measures, and gives a simple proof of the RudolphJohnson theorem. Our methods also apply to many other classes of conformal fractals and measures. As another application, we extend and unify Results of Peres, Shmerkin and Nazarov, and of Moreira, concerning projections of products selfsimilar measures and Gibbs measures on regular Cantor sets. We show that under natural irreducibility assumptions on the maps in the IFS, the image measure has the maximal possible dimension under any linear projection other than the coordinate projections. We also present applications to Bernoulli convolutions and to the images of fractal measures under differentiable maps.
 Publication:

arXiv eprints
 Pub Date:
 October 2009
 DOI:
 10.48550/arXiv.0910.1956
 arXiv:
 arXiv:0910.1956
 Bibcode:
 2009arXiv0910.1956H
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Classical Analysis and ODEs;
 28A80;
 28A78;
 37C45;
 37F35
 EPrint:
 55 pages. Version 2: Corrected an error in proof Thm. 4.3