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. Poincaré 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. Poincaré 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. Lüroth transformations and continued fractions; Exercises; Investigations; Notes; Bibliography; Index.
