Geodesic
On self-similar sets with overlaps and inverse theorems for entropy
Michael Hochman1
1 Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, Israel
Annals of mathematics, Tome 180 (2014) no. 2, pp. 773-822.

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We study the dimension of self-similar sets and measures on the line. We show that if the dimension is less than the generic bound of $\mathrm{min}\{1,s\}$, where $s$ is the similarity dimension, then there are superexponentially close cylinders at all small enough scales. This is a step towards the conjecture that such a dimension drop implies exact overlaps and confirms it when the generating similarities have algebraic coefficients. As applications we prove Furstenberg’s conjecture on projections of the one-dimensional Sierpinski gasket and achieve some progress on the Bernoulli convolutions problem and, more generally, on problems about parametric families of self-similar measures. The key tool is an inverse theorem on the structure of pairs of probability measures whose mean entropy at scale $2^{-n}$ has only a small amount of growth under convolution.
DOI : 10.4007/annals.2014.180.2.7

Michael Hochman 1

1 Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, Israel 
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Michael Hochman. On self-similar sets with overlaps and inverse theorems for entropy. Annals of mathematics, Tome 180 (2014) no. 2, pp. 773-822. doi : 10.4007/annals.2014.180.2.7. http://geodesic.mathdoc.fr/articles/10.4007/annals.2014.180.2.7/

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