Manfred Einsiedler Thomas Ward. Ergodic Theory. with a view towards Number Theory. ^ Springer

Transcription

1 Manfred Einsiedler Thomas Ward Ergodic Theory with a view towards Number Theory ^ Springer

2 1 Motivation Examples of Ergodic Behavior Equidistribution for Polynomials Szemeredi's Theorem Indefinite Quadratic Forms and Oppenheim's Conjecture Littlewood's Conjecture Integral Quadratic Forms Dynamics on Homogeneous Spaces An Overview of Ergodic Theory Ergodicity, Recurrence and Mixing Measure-Preserving Transformations Recurrence Ergodicity Associated Unitary Operators The Mean Ergodic Theorem Pointwise Ergodic Theorem The Maximal Ergodic Theorem Maximal Ergodic Theorem via Maximal Inequality Maximal Ergodic Theorem via a Covering Lemma The Pointwise Ergodic Theorem Two Proofs of the Pointwise Ergodic Theorem Strong-Mixing and Weak-Mixing Proof of Weak-Mixing Equivalences Continuous Spectrum and Weak-Mixing Induced Transformations 61 3 Continued Fractions Elementary Properties The Continued Fraction Map and the Gauss Measure 76 xiii

3 xiv 3.3 Badly Approximate Numbers Lagrange's Theorem Invertible Extension of the Continued Fraction Map 91 4 Invariant Measures for Continuous Maps Existence of Invariant Measures Ergodic Decomposition Unique Ergodicity Measure Rigidity and Equidistribution Equidistribution on the Interval Equidistribution and Generic Points Equidistribution for Irrational Polynomials Conditional Measures and Algebras Conditional Expectation Martingales Conditional Measures Algebras and Maps Factors and Joinings The Ergodic Theorem and Decomposition Revisited Invariant Algebras and Factor Maps The Set of Joinings Kronecker Systems Constructing Joinings Furstenberg's Proof of Szemeredi's Theorem Van der Waerden Multiple Recurrence Reduction to an Invertible System Reduction to Borel Probability Spaces Reduction to an Ergodic System Furstenberg Correspondence Principle An Instance of Polynomial Recurrence The van der Corput Lemma Two Special Cases of Multiple Recurrence Kronecker Systems Weak-Mixing Systems Roth's Theorem Proof of Theorem 7.14 for a Kronecker System Reducing the General Case to the Kronecker Factor Definitions Dichotomy Between Relatively Weak-Mixing and Compact Extensions 201

4 xv 7.9 SZ for Compact Extensions SZ for Compact Extensions via van der Waerden A Second Proof Chains of SZ Factors SZ for Relatively Weak-Mixing Extensions Concluding the Proof Further Results in Ergodic Ramsey Theory Other Furstenberg Ergodic Averages Actions of Locally Compact Groups Ergodicity and Mixing Mixing for Commuting Automorphisms Ledrappier's "Three Dots" Example Mixing Properties of the x2, x3 System Haar Measure and Regular Representation Measure-Theoretic 'Transitivity and Uniqueness Amenable Groups Definition of Amenability and Existence of Invariant Measures Mean Ergodic Theorem for Amenable Groups Pointwise Ergodic Theorems and Polynomial Growth Flows Pointwise Ergodic Theorems for a Class of Groups Ergodic Decomposition for Group Actions Stationary Measures Geodesic Flow on Quotients of the Hyperbolic Plane The Hyperbolic Plane and the Isometric Action The Geodesic Flow and the Horocycle Flow Closed Linear Groups. and Left Invariant Riemannian Metric The Exponential Map and the Lie Algebra of a Closed Linear Group The Left-Invariant Riemannian Metric Discrete Subgroups of Closed Linear Groups Dynamics on Quotients Hyperbolic Area and Fuchsian Groups Dynamics on r\psl2(m) Lattices in Closed Linear Groups Hopf's Argument for Ergodicity of the Geodesic Flow Ergodicity of the Gauss Map Invariant Measures and the Structure of Orbits Symbolic Coding Measures Coming from Orbits 328

5 xvi 10 Nilrotation Rotations on the Quotient of the Heisenberg Group The Nilrotation First Proof of Theorem Second Proof of Theorem A Commutative Lemma; The Set K Studying Divergence; The Set Xi Combining Linear Divergence and the Maximal Ergodic Theorem A Non-ergodic Nilrotation The General Nilrotation More Dynamics on Quotients of the Hyperbolic Plane Dirichlet Regions Examples of Lattices Arithmetic and Congruence Lattices in SL2(R) A Concrete Principal Congruence Lattice of SL2(R) Uniform Lattices Unitary Representations, Mautner Phenomenon, and Ergodicity Three Types of Actions Ergodicity Mautner Phenomenon for SL2(R) Mixing and the Howe-Moore Theorem First Proof of Theorem Vanishing of Matrix Coefficients for PSL2(R) Second Proof of Theorem 11.22; Mixing of All Orders Rigidity of Invariant Measures for the Horocycle Flow Existence of Periodic Orbits; Geometric Characterization Proof of Measure Rigidity for the Horocycle Flow Non-escape of Mass for Horocycle Orbits The Space of Lattices and the Proof of Theorem = forx2 SL2(Z)\SL2(R) Extension to the General Case Equidistribution of Horocycle Orbits 399 Appendix A: Measure Theory 403 A.l Measure Spaces 403 A.2 Product Spaces 406 A.3 Measurable Functions 407 A.4 Radon-Nikodym Derivatives 409 A.5 Convergence Theorems 410 A.6 Well-Behavecl Measure Spaces 411 A.7 Lebesgue Density Theorem 412 A.8 Substitution Rule 413

6 xvii Appendix B: Functional Analysis 417 B.l Sequence Spaces 417 B.2 Linear Functional 418 B.3 Linear Operators 419 B.4 Continuous Functions 421 B.5 Measures on Compact Metric Spaces 422 B.6 Measures on Other Spaces 425 B. 7 Vector-valued Integration 425 Appendix C: Topological Groups 429 C. l General Definitions 429 C.2 Haar Measure on Locally Compact Groups 431 C.3 Pontryagin Duality 433 Hints for Selected Exercises 441 References 447 Author Index 463 Index of Notation 467 General Index 471