Randomness and Recurrence in Dynamical Systems

Double-Click Image to Enlarge

Randomness and Recurrence in Dynamical Systems
Rodney Nillsen

Catalog Code: CAM-31 
ISBN: 978-0-88385-043-5 
357 pp., Hardbound, 2010 
List Price: $52.95 
Member Price: $42.95
Series: Carus Mathematical Monographs, #31

 

 

Table of Contents | Excerpt | About the Author |Buy on Amazon | Buy in MAA Bookstore

This book, part of the MAA's oldest book series, makes the ideas of randomness and recurrence in dynamical systems comprehensible for undergraduates and beginning graduate students. It fills the gap between undergraduate teaching and current mathematical research, bringing out relevant results with a minimum of measure theory.

Author Rodney Nillsen develops new techniques of proof and adapts known proofs to make the material accessible to students with a background only in elementary analysis.

Book Highlights
• Emphasizes interpretations of results, concepts, and connections to other areas of inquiry;
• Includes exercises, investigations, and more than 60 figures to explain proofs;
• Sets the mathematical ideas in historical context;
• Suggests areas for further study.

 

Table of Contents

Forward
Preface
Background Ideas and Knowledge

Dynamical systems, iteration, and orbits

Information loss and randomness in dynamical systems

Assumed knowledge and notations

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 Reich

The 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 approach

The 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 dynamical systems

Scale invariance and the necessity of Benford's

Exercises

Investigations

Notes

Bibliography

Recurrence

Introduction: random systems and recurrence

Transformations that preserve length

Poincaré recurrence

Recurrent points

Kac's result on average recurrence times

Applications to the Kronecker and Borel dynamical 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: the time average equals the space average

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 Kakutani

Lüroth transformations and continued fractions

Exercises

Investigations

Notes

Bibliography

Index of Subjects

 

Index of Symbols

Excerpt

The curious phenomenon of the leading significant digit (p. 180):

Suppose we have a collection of positive numbers, perhaps arising from a set of data. Assuming the data is random, we might expect that the leading digits of the numbers in the data would occur with an approximately equal frequency. So, it may come as a very surprising fact that this is often not the case. Back in the days when electronic calculators did not exist, arithmetical calculations were carried out using books of logarithmic tables. It seems to have been Simon Newcomb, the professor of mathematics and astronomy at Johns Hopkins University, who observed in 1881 that the pages near the front of books of logarithms were more used than the pages towards the back. He wrote:

"That the ten digits do not occur with equal frequency must be evident to any one making use of logarithmic tables, and noticing how much faster the first pages wear out than the last ones. The first significant digit is oftener 1 than any other digit, and the frequency diminishes up to 9."

About the Author

Rodney Nillsen (University of Wollongong, in New South Wales, Australia) received his undergraduate education at the University of Tasmania and postgraduate education at Flinders University of South Australia. A member of the MAA, he is interested in harmonic analysis, functional analysis, differential equations, and measure theory. He is the author of Differential Spaces and Invariant Linear Forms (1994).

 

Source : http://www.maa.org/pubs/CAM-31.html