Ergodic theory with a view towards number theory

Introduction......Page 7
Examples of Ergodic Behavior......Page 13
Equidistribution for Polynomials......Page 15
Szemerédi's Theorem......Page 16
Indefinite Quadratic Forms and Oppenheim's Conjecture......Page 17
Littlewood's Conjecture......Page 18
Integral Quadratic Forms......Page 19
Dynamics on Homogeneous Spaces......Page 21
An Overview of Ergodic Theory......Page 22
Measure-Preserving Transformations......Page 25
Recurrence......Page 32
Ergodicity......Page 34
Associated Unitary Operators......Page 40
The Mean Ergodic Theorem......Page 43
Pointwise Ergodic Theorem......Page 48
Strong-mixing and Weak-mixing......Page 59
Proof of Weak-mixing Equivalences......Page 64
Induced Transformations......Page 71
Elementary Properties......Page 79
The Continued Fraction Map and the Gauss Measure......Page 86
Badly Approximable Numbers......Page 96
Invertible Extension of the Continued Fraction Map......Page 101
Invariant Measures for Continuous Maps......Page 105
Existence of Invariant Measures......Page 106
Ergodic Decomposition......Page 111
Unique Ergodicity......Page 113
Measure Rigidity and Equidistribution......Page 118
Conditional Expectation......Page 129
Martingales......Page 133
Conditional Measures......Page 140
Algebras and Maps......Page 152
The Ergodic Theorem and Decomposition Revisited......Page 159
Invariant Algebras and Factor Maps......Page 162
The Set of Joinings......Page 164
Kronecker Systems......Page 165
Constructing Joinings......Page 169
Furstenberg's Proof of Szemerédi's Theorem......Page 177
Van der Waerden......Page 178
Multiple Recurrence......Page 181
Furstenberg Correspondence Principle......Page 184
An Instance of Polynomial Recurrence......Page 186
Two Special Cases of Multiple Recurrence......Page 193
Roth's Theorem......Page 197
Definitions......Page 203
Dichotomy Between Relatively Weak-mixing and Compact......Page 206
SZ for Compact Extensions......Page 212
Chains of SZ Factors......Page 220
SZ for Relatively Weak-Mixing Extensions......Page 222
Concluding the Proof......Page 229
Further Results in Ergodic Ramsey Theory......Page 230
Ergodicity and Mixing......Page 235
Mixing for Commuting Automorphisms......Page 239
Haar Measure and Regular Representation......Page 246
Amenable Groups......Page 254
Mean Ergodic Theorem for Amenable Groups......Page 258
Pointwise Ergodic Theorems and Polynomial Growth......Page 260
Ergodic Decomposition for Group Actions......Page 269
Stationary Measures......Page 275
The Hyperbolic Plane and the Isometric Action......Page 279
The Geodesic Flow and the Horocycle Flow......Page 284
Closed Linear Groups and a Left-Invariant Riemannian Metric......Page 290
Dynamics on Quotients......Page 306
Hopf's Argument for Ergodicity of the Geodesic Flow......Page 315
Ergodicity of the Gauss Map......Page 318
Invariant Measures and the Structure of Orbits......Page 328
Rotations on the Quotient of the Heisenberg Group......Page 333
The Nilrotation......Page 335
First Proof of Theorem 10.1......Page 336
Second Proof of Theorem 10.1......Page 337
A Non-ergodic Nilrotation......Page 342
The General Nilrotation......Page 344
Dirichlet Regions......Page 347
Examples of Lattices......Page 357
Unitary Representations, Mautner Phenomenon, Ergodicity......Page 363
Mixing and the Howe--Moore Theorem......Page 370
Rigidity of Invariant Measures for the Horocycle Flow......Page 377
Non-escape of Mass for Horocycle Orbits......Page 387
Equidistribution of Horocycle Orbits......Page 397
Measure Spaces......Page 401
Product Spaces......Page 404
Measurable Functions......Page 405
Radon--Nikodym Derivatives......Page 407
Convergence Theorems......Page 408
Well-behaved Measure Spaces......Page 409
Lebesgue Density Theorem......Page 410
Substitution Rule......Page 411
Sequence Spaces......Page 413
Linear Functionals......Page 414
Linear Operators......Page 415
Continuous Functions......Page 417
Measures on Compact Metric Spaces......Page 418
Measures on Other Spaces......Page 420
Vector-valued Integration......Page 421
General Definitions......Page 425
Haar Measure on Locally Compact Groups......Page 427
Pontryagin Duality......Page 429
Hints for Selected Exercises......Page 437
References......Page 441
General Index......Page 460