Local entropy averages and projections of fractal measures

By Michael Hochman, Pablo Shmerkin. 2009

189 citations recorded by the source corpus.

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: 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 times-m mod 1 and times-n mod 1. Then, for any non-zero t: 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 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.

Subfields: Computational Theory and Mathematics, Materials Chemistry, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/01a01c294132909a7fc56a06ddfae1332df3c684

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Citations in the corpus

A proof of Furstenberg's conjecture on the intersections of ⨯p- and ⨯q-invariant sets (2016, background): “[21] Hochman, Michael; Shmerkin, Pablo, Local entropy averages and projections of fractal measures, Ann.”
New Bounds on the Dimensions of Planar Distance Sets (2018, methodology): “We emphasize that this method to bound the dimension of (linear or nonlinear) projections from below goes back in various forms to [10, 9, 22], although the use of projected L norms (rather than projected entropies) was first used in [26].”
Sixty years of fractal projections (2014, methodology): “Shmerkin and Solomyak showed that this is very nearly so in the plane.”
Dimension of invariant measures for affine iterated function systems (2019, background): “[36] M. Hochman and P. Shmerkin.”
Equidistribution from fractal measures (2013, methodology): “4 which does not directly use EFDs and instead relies on the results from [27] on dissonance between quasi-product measures for regular IFSs.”
On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions (2013, background): “In the last few years, significant progress was achieved in verifying that the exceptional set in the sub-critical regime is indeed very small (countable or zerodimensional) for many important fractal families, including convolutions of central Cantor measures [17], convolutions of ×p-invariant meas”
On Furstenberg's intersection conjecture, self-similar measures, and the Lq norms of convolutions (2016, background): “Moreover, in [28] a corresponding result for invariant measures was obtained: if μ, ν are Borel probability measures invariant under ×p,×q respectively, then”
Absolute continuity of self-similar measures, their projections and convolutions (2014, methodology): “For this we apply, depending on the context, any of a number of recent results on the dimension of measures of dynamical origin [11, 13, 18].”