Local entropy averages and projections of fractal measures

Michael Hochman*, Pablo Shmerkin

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

100 Scopus citations

Abstract

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 ⊆ [0,1] are closed and invariant, respectively, under ×m mod 1 and ×n mod 1, where m, n are not powers of the same integer, then, for any t ≠ 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.

Original languageAmerican English
Pages (from-to)1001-1059
Number of pages59
JournalAnnals of Mathematics
Volume175
Issue number3
DOIs
StatePublished - May 2012

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