Journal of Modern Dynamics (JMD)

Lectures on dynamics, fractal geometry, and metric number theory

Pages: 437 - 497, Issue 3/4, September/December 2014      doi:10.3934/jmd.2014.8.437

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Michael Hochman - Einstein Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel (email)

Abstract: 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\rightarrow ax $mod1 and $x\mapsto bx$mod1 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.

Keywords:  Hausdorff dimension, entropy dimension, Marstrand projection theorem, Marstrand slice theorem, CP-process, transversality of semigroups.
Mathematics Subject Classification:  Primary: 28A80; Secondary: 37F99, 37A99.

Received: December 2013;      Revised: August 2014;      Available Online: April 2015.