Ergodic Theory and Differentiable Dynamics

0. Measure Theory.- 1. Measures.- 2. Measurable Maps.- 3. Integrable Functions.- 4. Differentiation and Integration.- 5. Partitions and Derivatives.- I. Measure-Preserving Maps.- 1. Introduction.- 2. The Poincare Recurrence Theorem.- 3. Volume-Preserving Diffeomorphisms and Flows.- 4. First Integrals.- 5. Hamiltonians.- 6. Continued Fractions.- 7. Topological Groups, Lie Groups, Haar Measure.- 8. Invariant Measures.- 9. Uniquely Ergodic Maps.- 10. Shifts: the Probabilistic Viewpoint.- 11. Shifts: the Topological Viewpoint.- 12. Equivalent Maps.- II. Ergodicity.- 1. Birkhoff's Theorem.- 2. Ergodicity.- 3. Ergodicity of Homomorphisms and Translations of the Torus.- 4. More Examples of Ergodic Maps.- 5. The Theorem of Kolmogorov-Arnold-Moser.- 6. Ergodic Decomposition of Invariant Measures.- 7. Furstenberg's Example.- 8. Mixing Automorphisms and Lebesgue Automorphisms.- 9. Spectral Theory.- 10. Gaussian Shifts.- 11. Kolmogorov Automorphisms.- 12. Mixing and Ergodic Markov Shifts.- III. Expanding Maps and Anosov Diffeomorphisms.- 1. Expanding Maps.- 2. Anosov Diffeomorphisms.- 3. Absolute Continuity of the Stable Foliation.- IV. Entropy.- 1. Introduction.- 2. Proof of the Shannon-McMillan-Breiman Theorem.- 3. Entropy.- 4. The Kolmogorov-Sinai Theorem.- 5. Entropy of Expanding Maps.- 6. The Parry Measure.- 7. Topological Entropy.- 8. The Variational Property of Entropy.- 9. Hyperbolic Homeomorphisms.- 10. Lyapunov Exponents. The Theorems of Oseledec and Pesin.- 11. Proof of Oseledec's Theorem.- 12. Proof of Ruelle's Inequality.- 13. Proof of Pesin's Formula.- 14. Entropy of Anosov Diffeomorphisms.- 15. Hyperbolic Measures. Katok's Theorem.- 16. The Brin-Katok Local Entropy Formula.- Notation Index.