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

• Autores: Michael Hochman
• Localización: Annals of mathematics, ISSN 0003-486X, Vol. 180, Nº 2, 2014, págs. 773-822
• Idioma: inglés
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• Resumen

  • 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 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