A First Course in Ergodic Theory

Preface. Author Bios. 1. Measure preservingness and basic examples. 1.1. What is Ergodic Theory. 1.2. Measure Preserving Transformations. 1.3. Basic Examples. 2. Recurrence and Ergodicity. 2.1. Recurrence. 2.2. Ergodicity. 2.3. Examples of Ergodic Transformations. 3. The Pointwise Ergodic Theorem and its consequences. 3.2. Normal Numbers. 3.3. Characterization of Irreducible Markov Chains. 3.4. Mixing. 4. More Ergodic Theorem. The mean Ergodic Theorem. 4.2. The Hurewicz Erogdic Theorem. 5. Measure Preserving Isomorphisms. 5.2. Factor Maps. 5.3. Natural Extensions. 6. The Perron-Frobenius Operator. 6.1. Absolutely Continuous Invariants Measures. 6.2. Exactness. Densities for Piecewise Monotnoe Interval Maps. 7. Invariant Measures for Continuous Transformations. 7.1. Existence. 7.2. Unique Ergodicity and Inform Distributions. 7.3. Some Topological Dynamics. 8. Continued Fractions. 8.1. Basic Properties of Regular Continue Fractions. 8.2. Ergodic Properties of Gauss Map. 8.3. Natural Extension and the Doeblin-Lenstra Conjecture. 8.4. Other Continue Fraction Transformation. 9. Entropy. 9.1. Randomness and Information. 9.2. Definitions and Properties. Calculation of Entropy and Examples. 9.4. The Shannon-McMillan-Breiman Theorem. 9.5. Lochs' Theorem. 10. The Variational Principle. 10.1 Topological Entropy. 10.2. Main Theorem. 10.3. Measures of Maximal Entropy. 11. Infinite Ergodic Theory. 11.1 Examples of Infinite Measure Dynamical Systems. 11.2. Conservative and Dissipative Part. 11.3. Induced Systems. 11.4. Jump Transformations. 11.5. Ergodic Theorem for Infinite Measure Systems. 12. Appendix. 12.1. Topology. 12.2. Measure Theory. 12.3 Lebesgue Spaces. 12.4. Lebesgue Integration and Convergence Results. 12.5. Hilbert's Spaces. 12.6. Borel Measures on Compact Metric Spaces. 12.7. Functions of Bounded Variation. Bibliography. Index.