TY - JOUR

T1 - Local entropy averages and projections of fractal measures

AU - Hochman, Michael

AU - Shmerkin, Pablo

PY - 2012/5

Y1 - 2012/5

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84861581243&partnerID=8YFLogxK

U2 - 10.4007/annals.2012.175.3.1

DO - 10.4007/annals.2012.175.3.1

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AN - SCOPUS:84861581243

SN - 0003-486X

VL - 175

SP - 1001

EP - 1059

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 3

ER -