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Fractals, algebraic dynamics and metric number theory (FRACTALSANDMETRICNT)
Date du début: 1 oct. 2012, Date de fin: 30 sept. 2017 PROJET  TERMINÉ 

We propose to study the fractal geometry of invariant sets for endomorphisms of compact abelian groups, specifically a family of conjectures by Furstenberg on the dimensions of orbit closures under such dynamics, and on the size of sums and intersections of invariant sets. These conjectures are related to problems on expansion in integer bases, in Diophantine approximation, measure rigidity, analysis and equidistribution. The project focuses on the conjectures themselves and some related problems, e.g. Bernoulli convolutions, and on applications to equidistribution on tori. Our approach combines tools from ergodic theory, geometric measure theory and additive combinatorics, building on recent progress in these fields and recent partial results towards the main conjectures.