Randomness and Recurrence in Dynamical Systems: A Real Analysis Approach: 31 - Rilegato

Nillsen, Rodney

Rilegato

ISBN 10: 0883850435 ISBN 13: 9780883850435

Casa editrice: The Mathematical Association of America, 2010

Riassunto

An accessible introduction to real analysis for undergraduates, focussing on applications of analysis to abstract dynamical systems.

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Descrizione del libro

This work is an accessible introduction to real analysis as it applies to topics in abstract dynamical systems for undergraduates which presupposes little advanced knowledge. It connects its results to recent research and thus aims to bridge the gap between undergraduate teaching and current mathematical ideas and research.

Contenuti

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.

Dettagli dell'edizione

• Editore: The Mathematical Association of America
• Data di pubblicazione: 2010
• Lingua: Inglese
• ISBN 10: 0883850435
• ISBN 13: 9780883850435
• Rilegatura: Copertina rigida
• Numero di pagine: 374
• Contatto del produttore: non disponibile
• Persona responsabile: non disponibile