On self-similar sets with overlaps and inverse theorems for entropy

Michael Hochman*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

155 Scopus citations

Abstract

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 minf1; sg, 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 coeffcients. 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.

Original languageAmerican English
Pages (from-to)773-822
Number of pages50
JournalAnnals of Mathematics
Volume180
Issue number2
DOIs
StatePublished - 2014

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