LECTURES ON DYNAMICS, FRACTAL GEOMETRY, AND METRIC NUMBER THEORY

Michael Hochman*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

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 → 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.

Original languageAmerican English
Pages (from-to)437-497
Number of pages61
JournalJournal of Modern Dynamics
Volume8
Issue number3-4
DOIs
StatePublished - 2014

Bibliographical note

Publisher Copyright:
© 2014 AIMSCIENCES.

Keywords

  • CP-process
  • Entropy dimension
  • Hausdorff dimension
  • Marstrand projection theorem
  • Marstrand slice theorem
  • Transversality of semigroups

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