Presenting a wide cross-section of current research in the theory of dynamical systems, this collection consists of articles by leading researchers (and several Fields medallists) in a variety of specialties. Surveys featuring new results, as well as research papers, are included because of their potentially high impact. Major areas covered include hyperbolic dynamics, elliptic dynamics, mechanics, geometry, ergodic theory, group actions, rigidity, and applications. The target audience is dynamicists, as well as mathematicians from other disciplines.

### Contents

1. Introduction Michael Brin, Boris Hasselblatt and Yakov Pesin; Part I. Ergodic Theory, Rigidity, Geometry: 2. Weakly mixing actions of general groups: a brief survey and an example Vitaly Bergelson and Alex Gorodnik; 3. Dynamical Morse entropy Melanie Bertelson and Michael Gromov; 4. Positive k-theory and symbolic dynamics Michael Boyle and Jack Wagoner; 5. Geometry of 2-step nilpotent Lie groups with a left invariant metric Patrick Eberlein; 6. A differential-geometric view of normal forms of contractions Renato Feres; 7. Averaging along cubes B. Host and Bryna Kra; 8. Sections for semiflows and Kakutani shift equivalence Chao-Hui Lin and Daniel Rudolph; 9. Coarsely geodesic metrics on reductive groups Herbert Abels and Gregory Margulis; 10. Algebraic Zd-actions on zero-dimensional compact Abelian groups Klaus Schmidt; 11. An invitation to rigidity theory Ralf Spatzier; 12. Remarks on magnetic flows and magnetic billiards, Finsler metrics and a magnetic analog of Hilbert's fourth problem Serge Tabachnikov; Part II. Hyperbolic Dynamics: 13. Expanding polymodials Alexander Blokh, Chris Cleveland and Michal Misiurewicz; 14. Lyapunov exponents: how frequently are dynamical systems hyperbolic? Jairo Bochi and Marcelo Viana; 15. A HÃ¶lder continuous vector field tangent to many foliations Christina Bonatti and John Franks; 16. On partially hyperbolic diffeomorphisms of 3-manifolds with commutative fundamental groups Michael Brin, Dmitri Burago and S. Ivanov; 17. Prelude to a kiss Dmitry Dolgopyat; 18. Nonuniform hpyerbolicity and elliptic dynamics Bassam Fayad; 19. Dimension product structure of hyperbolic sets Boris Hasselblatt and Jorg Schmeling; 20. Every compact manifold carries a hyperbolic Bernoulli flow Huyi Hu, Yakov Pesin and Anna Talitskaya; 21. Parameter choice for families of maps with many critical points Michael Jakobson; 22. Entropy, exponents and invariant densities for hyperbolic systems: dependence and computation Oliver Jenkinson and Mark Pollicott; 23. Some recent advances in averaging Yuri Kifer; 24. Bootstrap of regularity for integrable solutions of cohomology equations Rafael de la Llave; 25. Cone-fields, domination, and hyperbolicity Sheldon Newhouse; 26. Markov towers and stochastic properties of billiards Domokos Szasz and Tamas Varju; 27. Some questions and remarks about SL(2,R) cocycles Jean-Christophe Yoccoz.