Randomness and Recurrence in Dynamical Systems: A Real Analysis Approach (Carus Mathematical Monographs S.)
By: Rodney Nillsen (author)Hardback
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Randomness and Recurrence in Dynamical Systems makes accessible, at the undergraduate or beginning graduate level, results and ideas on averaging, randomness and recurrence that traditionally require measure theory. Assuming only a background in elementary calculus and real analysis, new techniques of proof have been developed, and known proofs have been adapted, to make this possible. The book connects the material with recent research, thereby bridging the gap between undergraduate teaching and current mathematical research. The various topics are unified by the concept of an abstract dynamical system, so there are close connections with what may be termed 'Probabilistic Chaos Theory' or 'Randomness'. The work is appropriate for undergraduate courses in real analysis, dynamical systems, random and chaotic phenomena and probability. It will also be suitable for readers who are interested in mathematical ideas of randomness and recurrence, but who have no measure theory background.
Introduction: 1. Origins, approach and aims of the work; 2. Dynamical systems and the subject matter; 3. Using this book; Part I. Background Ideas and Knowledge: 4. Dynamical systems, iteration, and orbits; 5. Information loss and randomness in dynamical systems; 6. Assumed knowledge and notations; Appendix: mathematical reasoning and proof; Exercises; Investigations; Notes; Bibliography; Part II. Irrational Numbers and Dynamical Systems: 7. Introduction: irrational numbers and the infinite; 8. Fractional parts and points on the unit circle; 9. Partitions and the pigeon-hole principle; 10. Kronecker's theorem; 11. The dynamical systems approach to Kronecker's theorem; 12. Kronecker and chaos in the music of Steve Reich; 13. The ideas in Weyl's theorem on irrational numbers; 14. The proof of Weyl's theorem; 15. Chaos in Kronecker systems; Exercises; Investigations; Notes; Bibliography; Part III. Probability and Randomness: 16. Introduction: probability, coin tossing and randomness; 17. Expansions to a base; 18. Rational numbers and periodic expansions; 19. Sets, events, length and probability; 20. Sets of measure zero; 21. Independent sets and events; 22. Typewriters, recurrence, and the Prince of Denmark; 23. The Rademacher functions; 24. Randomness, binary expansions and a law of averages; 25. The dynamical systems approach; 26. The Walsh functions; 27. Normal numbers and randomness; 28. Notions of probability and randomness; 29. The curious phenomenon of the leading significant digit; 30. Leading digits and geometric sequences; 31. Multiple digits and a result of Diaconis; 32. Dynamical systems and changes of scale; 33. The equivalence of Kronecker and Benford dynamical systems; 34. Scale invariance and the necessity of Benford's law; Exercises; Investigations; Notes; Bibliography; Part IV. Recurrence: 35. Introduction: random systems and recurrence; 36. Transformations that preserve length; 37. Poincare recurrence; 38. Recurrent points; 39. Kac's result on average recurrence times; 40. Applications to the Kronecker and Borel dynamical systems; 41. The standard deviation of recurrence times; Exercises; Investigations; Notes; Bibliography; Part V. Averaging in Time and Space: 42. Introduction: averaging in time and space; 43. Outer measure; 44. Invariant sets; 45. Measurable sets; 46. Measure-preserving transformations; 47. Poincare recurrence ... again!; 48. Ergodic systems; 49. Birkhoff's theorem: the time average equals the space average; 50. Weyl's theorem from the ergodic viewpoint; 51. The Ergodic Theorem and expansions to an arbitrary base; 52. Kac's recurrence formula: the general case; 53. Mixing transformations and an example of Kakutani; 54. Luroth transformations and continued fractions; Exercises; Investigations; Notes; Bibliography; Index.
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