Local entropy averages and projections of fractal measures

TL;DR

This paper investigates how the dimension of linear images of self-similar measures in Euclidean space behaves semi-continuously, proving a conjecture related to invariant fractal sets and extending results on projections of fractal measures.

Contribution

It introduces a new approach using local entropy averages for self-similar measures, proving Furstenberg's conjecture and extending projection theorems under irreducibility conditions.

Findings

Proves Furstenberg's conjecture on dimensions of sumsets of invariant fractals.

Shows maximal dimension preservation under linear projections for certain self-similar measures.

Extends results to Bernoulli convolutions and differentiable images of fractal measures.

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: 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.