Local entropy averages and projections of fractal measures

@article{Hochman2012LocalEA,
  title={Local entropy averages and projections of fractal measures},
  author={M. Hochman and P. Shmerkin},
  journal={Annals of Mathematics},
  year={2012},
  volume={175},
  pages={1001-1059}
}
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… Expand

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