This second half of Volume 1 of this Handbook follows Volume 1A, which was published in 2002. The contents of these two tightly integrated parts taken together come close to a realization of the program formulated in the introductory survey "Principal Structures" of Volume 1A. The present volume contains surveys on subjects in four areas of dynamical systems: Hyperbolic dynamics, parabolic dynamics, ergodic theory and infinite-dimensional dynamical systems (partial differential equations). . Written by experts in the field. . The coverage of ergodic theory in these two parts of Volume 1 is considerably more broad and thorough than that provided in other existing sources. . The final cluster of chapters discusses partial differential equations from the point of view of dynamical systems.
Author(s): A. Katok, B. Hasselblatt
Series: Handbook of Dynamical Systems
Publisher: Elsevier Science & Technology
Year: 1997
Language: English
Pages: 1235
Preface......Page 6
List of Contributors......Page 8
Contents......Page 10
Contents of Volume 1A......Page 12
Partially Hyperbolic Dynamical Systems......Page 14
Introduction......Page 16
Definitions and examples......Page 20
Filtrations of stable and unstable foliations......Page 30
Central foliations......Page 34
Intermediate foliations......Page 40
Failure of absolute continuity......Page 44
Accessibility and stable accessibility......Page 47
The Pugh-Shub ergodicity theory......Page 54
Partially hyperbolic attractors......Page 61
References......Page 65
Smooth Ergodic Theory and Nonuniformly Hyperbolic Dynamics......Page 70
Introduction......Page 74
Lyapunov exponents of dynamical systems......Page 75
Examples of systems with nonzero exponents......Page 79
Lyapunov exponents associated with sequences of matrices......Page 93
Cocycles and Lyapunov exponents......Page 100
Regularity and Multiplicative Ergodic Theorem......Page 110
Cocycles over smooth dynamical systems......Page 129
Methods for estimating exponents......Page 137
Local manifold theory......Page 147
Global manifold theory......Page 162
Absolute continuity......Page 168
Smooth invariant measures......Page 173
Metric entropy......Page 188
Genericity of systems with nonzero exponents......Page 196
SRB-measures......Page 209
Hyperbolic measures I: Topological properties......Page 218
Hyperbolic measures II: Entropy and dimension......Page 227
Geodesic flows on manifolds without conjugate points......Page 234
Dynamical systems with singularities: The conservative case......Page 240
Hyperbolic attractors with singularities......Page 245
Appendix A. Decay of correlations, by Omri Sarig......Page 257
References......Page 267
Stochastic-Like Behaviour in Nonuniformly Expanding Maps......Page 278
Introduction......Page 280
Basic definitions......Page 282
Markov structures......Page 288
Uniformly expanding maps......Page 298
Almost uniformly expanding maps......Page 300
One-dimensional maps with critical points......Page 302
General theory of nonuniformly expanding maps......Page 314
Existence of nonuniformly expanding maps......Page 317
Conclusion......Page 328
References......Page 333
Homoclinic Bifurcations, Dominated Splitting, and Robust Transitivity......Page 340
Introduction......Page 342
A weaker form of hyperbolicity: Dominated splitting......Page 343
Homoclinic tangencies......Page 347
Surface diffeomorphisms......Page 350
Nonhyperbolic robustly transitive systems......Page 366
Flows and singular splitting......Page 381
References......Page 387
Random Dynamics......Page 392
Introduction......Page 394
Basic structures of random transformations......Page 396
Smooth RDS: Invariant manifolds......Page 412
Relations between entropy, exponents and dimension......Page 430
Thermodynamic formalism and its applications......Page 457
Random perturbations of dynamical systems......Page 488
Concluding remarks......Page 503
References......Page 507
An Introduction to Veech Surfaces......Page 514
Introduction to Veech surfaces......Page 516
State of the art......Page 526
References......Page 537
Ergodic Theory of Translation Surfaces......Page 540
Three definitions of translation surface or flat surface and examples......Page 542
Spaces of translations surfaces and Riemann surfaces......Page 546
SL(2,R)-action and invariant measures......Page 547
Ergodicity of flows defined by translation surfaces......Page 549
Further results on unique ergodicity......Page 553
Boshernitzan's Theorem and sketch of proof of Theorem 3......Page 555
Further results on dynamics of actions of subgroups of SL(2,R)......Page 557
References......Page 559
On the Lyapunov Exponents of the Kontsevich-Zorich Cocycle......Page 562
Introduction......Page 564
Elements of Teichmüller theory......Page 567
The Kontsevich-Zorich cocycle......Page 571
Variational formulas......Page 573
Bounds on the exponents......Page 577
The determinant locus......Page 579
An example......Page 583
Invariant sub-bundles......Page 586
References......Page 591
Counting Problems in Moduli Space......Page 594
LECTURE 1: Counting problems and volumes of strata......Page 596
LECTURE 2: Lattice points and branched covers......Page 599
LECTURE 3: The Oppenheim conjecture and Ratner's theorem......Page 602
References......Page 607
On the Interplay between Measurable and Topological Dynamics......Page 610
Introduction......Page 612
Poincaré recurrence vs. Birkhoff's recurrence......Page 613
The equivalence of weak mixing and continuous spectrum......Page 618
Disjointness: measure vs. topological......Page 621
Mild mixing: measure vs. topological......Page 622
Distal systems: topological vs. measure......Page 630
Furstenberg-Zimmer structure theorem vs. its topological PI version......Page 632
Entropy: measure and topological......Page 634
Unique ergodicity......Page 646
The relative Jewett-Krieger theorem......Page 647
Models for other commutative diagrams......Page 653
Cantor minimal representations......Page 654
Other related theorems......Page 655
References......Page 658
Spectral Properties and Combinatorial Constructions in Ergodic Theory......Page 662
Spectral theory for Abelian groups of unitary operators......Page 664
Spectral properties and typical behavior in ergodic theory......Page 675
General properties of spectra......Page 684
Some aspects of theory of joinings......Page 697
Combinatorial constructions and applications......Page 704
Key examples outside combinatorial constructions......Page 741
Acknowledgements......Page 750
References......Page 751
Combinatorial and Diophantine Applications of Ergodic Theory......Page 758
Introduction......Page 760
Topological dynamics and partition Ramsey theory......Page 775
Dynamical, combinatorial, and Diophantine applications of betaN......Page 790
Multiple recurrence......Page 806
Actions of amenable groups......Page 838
Issues of convergence......Page 851
Appendix A. Host-Kra and Ziegler factors and convergence of multiple ergodic averages, by A. Leibman......Page 854
Appendix B. Ergodic averages along the squares, by A. Quas and M. Wierdl......Page 866
References......Page 877
Pointwise Ergodic Theorems for Actions of Groups......Page 884
Introduction......Page 886
Averaging along orbits in group actions......Page 888
Ergodic theorems for commutative groups......Page 892
Invariant metrics, volume growth, and ball averages......Page 896
Pointwise ergodic theorems for groups of polynomial volume growth......Page 906
Amenable groups: Følner averages and their applications......Page 914
A non-commutative generalization of Wiener's theorem......Page 922
Spherical averages......Page 936
The spectral approach to maximal inequalities......Page 944
Groups with commutative radial convolution structure......Page 953
Actions with a spectral gap......Page 966
Beyond radial averages......Page 976
Weighted averages on discrete groups and Markov operators......Page 982
Further developments......Page 986
References......Page 990
Global Attractors in PDE......Page 996
Introduction......Page 998
Global attractors of semigroups......Page 1002
Properties of attractors......Page 1024
Dynamical systems in function spaces......Page 1033
Generalized attractors......Page 1072
References......Page 1082
Hamiltonian PDEs......Page 1100
Symplectic Hilbert scales and Hamiltonian equations......Page 1102
Basic theorems on Hamiltonian systems......Page 1108
Lax-integrable equations......Page 1110
KAM for PDEs......Page 1114
Around the Nekhoroshev theorem......Page 1126
Invariant Gibbs measures......Page 1128
The non-squeezing phenomenon and symplectic capacity......Page 1129
The squeezing phenomenon and the essential part of the phase-space......Page 1134
Acknowledgements......Page 1136
Appendix. Families of periodic orbits in reversible PDEs, by D. Bambusi......Page 1137
References......Page 1144
Extended Hamiltonian Systemsextended Hamiltonian systems......Page 1148
Overview......Page 1150
Linear and nonlinear bound states......Page 1152
Orbital stability of ground states......Page 1154
Asymptotic stability of ground states I. No neutral oscillations......Page 1156
Resonance and radiation damping of neutral oscillations-metastability of bound states of the nonlinear Klein-Gordon equation......Page 1157
Acknowledgements......Page 1159
References......Page 1164
Author Index of Volume 1A......Page 1168
Subject Index of Volume 1A......Page 1182
Author Index......Page 1200
Subject Index......Page 1218
