Inhalt: | Preface -- List of Contributors -- Contents of Volume 1A -- 1. Partially Hyperbolic Dynamical Systems (B. Hasselblatt and Ya. Pesin) -- 2. Smooth Ergodic Theory and Nonuniformly Hypoerbolic Dynamics (L. Barreira and Ya. Pesin, with an Appendix by O. Sarig) -- 3. Stochastic-Like Behaviour in Nonuniformly Expanding Maps (S. Luzzatto) -- 4. Homoclinic Bifurcations, Dominated Splitting, and Robust Transivity (E.R. Pujals and M. Sambarino) -- 5. Random Dynamics (Yu. Kifer, P.-D. Liu) -- 6. An Introduction to Veech Surfaces (P. Hubert and T.A. Schmidt) -- 7. Ergodic Theory of Translation Surfaces (H. Masur) -- 8. On the Lyapunov Exponents of the Kontsevich-Zorich Cocycle (G. Forni) -- 9. Counting Problems in Moduli Space (A. Eskin) -- 10. On the Interplay Between Measurable and Topological Dynamics (E. Glasner and B. Weiss) -- 11. Spectral Properties and Combinatorial Constructions in Ergodic Theory (A. Katok and J.-P. Thouvenot) -- 12. Combinatorial and Diophantine Applications of Ergodic Theory (V. Bergelson, with Appendix A by A. Leibman and Appendix B by A. Quas and M. Wierdl) -- 13. Pointwise Ergodic Theorems for Actions of Groups (A. Nevo) -- 14. Global Attractors in PDE (A.V. Babin) -- 15. Hamiltonian PDEs (S.B. Kuksin, with an Appendix by D. Bambusi) -- 16. Extended Hamiltonian Systems (M.I. Weinstein) -- Author Index of Volume 1A -- Subject Index of Volume 1A -- Author Index -- Subject Index This handbook is volume II in a series collecting mathematical state-of-the-art surveys in the field of dynamical systems. Much of this field has developed from interactions with other areas of science, and this volume shows how concepts of dynamical systems further the understanding of mathematical issues that arise in applications. Although modeling issues are addressed, the central theme is the mathematically rigorous investigation of the resulting differential equations and their dynamic behavior. However, the authors and editors have made an effort to ensure readability on a non-technical level for mathematicians from other fields and for other scientists and engineers. The eighteen surveys collected here do not aspire to encyclopedic completeness, but present selected paradigms. The surveys are grouped into those emphasizing finite-dimensional methods, numerics, topological methods, and partial differential equations. Application areas include the dynamics of neural networks, fluid flows, nonlinear optics, and many others. While the survey articles can be read independently, they deeply share recurrent themes from dynamical systems. Attractors, bifurcations, center manifolds, dimension reduction, ergodicity, homoclinicity, hyperbolicity, invariant and inertial manifolds, normal forms, recurrence, shift dynamics, stability, to name just a few, are ubiquitous dynamical concepts throughout the articles |
