Local entropy averages and projections of fractal measures

• 成果类型: Article
• 署名作者: Hochman, Michael; Shmerkin, Pablo
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2012.175.3.1
• 发表日期: 2012
• 页码: 1001-1059
• 关键词: tangent measure distributions; hyperbolic cantor sets; dimension; scenery; convolutions; invariant; circle

摘要

We show that for families of measures on Euclidean space which satisfy an ergodic-theoretic form of self-similarity under the operation of re-scaling, the dimension of linear images of the measure behaves in a semi-continuous way. We apply this to prove the following conjecture of Furstenberg: if X, Y subset of [0, 1] are closed and invariant, respectively, under xm mod 1 and xn mod 1, where m, n are not powers of the same integer, then, for any t not equal 0, dim(X + tY) = min{1, dim X + dim Y}. A similar result holds for invariant measures and gives a simple proof of the Rudolph-Johnson 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 of self-similar 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.