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Randomness and Recurrence in Dynamical Systems : a Real Analysis Approach /
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...
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Cambridge :
Cambridge University Press,
2011.
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| Colección: | Carus.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Copyright page ; Title page ; Contents; Foreword; Preface; Background Ideas and Knowledge; Dynamical systems, iteration and orbits; Information loss and randomness in dynamical systems; Assumed knowledge and notation; Appendix: Mathematical reasoning and proof; Exercises; Investigations; Notes; Bibliography; Irrational Numbers and Dynamical Systems ; Introduction: irrational numbers and the infinite; Fractional parts and points on the unit circle; Partitions and the Pigeon-hole Principle ; Kronecker's Theorem; The dynamical systems approach to Kronecker's Theorem.
- Kronecker and chaos in the music of Steve ReichThe ideas in Weyl's Theorem on irrational numbers; The proof of Weyl's Theorem; Chaos in Kronecker systems; Exercises; Investigations; Notes; Bibliography; Probability and Randomness; Introduction: probability, coin tossing and randomness; Expansions to a base; Rational numbers and periodic expansions; Sets, events, length and probability; Sets of measure zero; Independent sets and events; Typewriters, recurrence, and the Prince of Denmark; The Rademacher functions; Randomness, binary expansions and a law of averages.
- The dynamical systems approachThe Walsh functions; Normal numbers and randomness; Notions of probability and randomness; The curious phenomenon of the leading significant digit; Leading digits and geometric sequences; Multiple digits and a result of Diaconis; Dynamical systems and changes of scale; The equivalence of Kronecker and Benford systems; Scale invariance and the necessity of Benford's law; Exercises; Investigations; Notes; Bibliography; Recurrence; Introduction: random systems and recurrence; Transformations that preserve length; Poincaré recurrence; Recurrent points.
- Kac's result on average recurrence timesApplications to the Kronecker and Borel systems; The standard deviation of recurrence times; Exercises; Investigations; Notes; Bibliography; Averaging in Time and Space; Introduction: averaging in time and space; Outer measure; Invariant sets; Measurable sets; Measure-preserving transformations; Poincaré recurrence ... again; Ergodic systems; Birkhoff's Theorem on time and space averages; Weyl's Theorem from the ergodic viewpoint; The Ergodic Theorem and expansions to an arbitrary base; Kac's recurrence formula: the general case.
- Mixing transformations and an example of KakutaniLüroth transformations and continued fractions; Exercises; Investigations; Notes; Bibliography; Bibliography; Index of Subjects; Index of Symbols; About the Author.