LECTURES ON DYNAMICS, FRACTAL GEOMETRY, AND METRIC NUMBER THEORY

These notes are based on lectures delivered in the summer school “Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory”, held in Będlewo, Poland, in the summer of 2011. The course is an exposition of Furstenberg’s conjectures on “transversality” of the maps x → ax mod 1 and x → bx mod 1 for multiplicatively independent integers a,b, and of the associated problems on intersections and sums of invariant sets for these maps. The first part of the course is a short introduction to fractal geometry. The second part develops the theory of Furstenberg’s CP-chains and local entropy averages, ending in proofs of the sumset problem and of the known case of the intersections conjecture.