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Randomness and Recurrence in Dynamical Systems

Randomness and Recurrence in Dynamical Systems
A Real Analysis Approach


Part of Carus Mathematical Monographs

  • Date Published: November 2010
  • availability: This item is not supplied by Cambridge University Press in your region. Please contact Mathematical Association of America for availability.
  • format: Hardback
  • isbn: 9780883850435

£ 39.99

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About the Authors
  • 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.

    • An emphasis on possible interpretations of certain results and concepts, and their connections to other areas of inquiry, gives the reader a depth of understanding
    • Includes both 'Exercises' and 'Investigations': the former emphasise more technical questions concerning the ideas, while the latter are more open, allowing scope for student initiative and further research
    • Notes at the end of each part set the mathematical ideas in their historical background
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    Product details

    • Date Published: November 2010
    • format: Hardback
    • isbn: 9780883850435
    • length: 374 pages
    • dimensions: 215 x 143 x 25 mm
    • weight: 0.54kg
    • availability: This item is not supplied by Cambridge University Press in your region. Please contact Mathematical Association of America for availability.
  • Table of Contents

    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
    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
    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
    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
    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

  • Author

    Rodney Nillsen

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