Author | Maรฑรฉ, Ricardo. author |
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Title | Ergodic Theory and Differentiable Dynamics [electronic resource] / by Ricardo Maรฑรฉ |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1987 |
Connect to | http://dx.doi.org/10.1007/978-3-642-70335-5 |
Descript | XII, 319 p. online resource |
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 Poincarรฉ 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. Birkhoffโs 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. Furstenbergโs 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 Oseledecโs Theorem -- 12. Proof of Ruelleโs Inequality -- 13. Proof of Pesinโs Formula -- 14. Entropy of Anosov Diffeomorphisms -- 15. Hyperbolic Measures. Katokโs Theorem -- 16. The Brin-Katok Local Entropy Formula -- Notation Index