Randomness and Recurrence in Dynamical Systems
A Real Analysis Approach

Table of Contents

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.