Handbook of Dynamical Systems (eBook)

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