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

**1. Background Ideas and Knowledge**

1.1 Dynamical systems, iteration, and orbits

1.2 Information loss and randomness in dynamical systems

1.3 Assumed knowledge and notations

Appendix: Mathematical reasoning and proof

Exercises

Investigations

Notes

Bibliography

**2. Irrational Numbers and Dynamical Systems**

2.1 Introduction: irrational numbers and the infinite

2.2 Fractional parts and points on the unit circle

2.3 Partitions and the pigeon-hole principle

2.4 Kronecker's Theorem

2.5 The dynamical systems approach to Kronecker's Theorem

2.6 Kronecker and chaos in the music of Steve Reich

2.7 The ideas in Weyl's theorem on irrational numbers

2.8 The proof of Weyl's theorem

2.9 Chaos in Kronecker systems

Exercises

Investigations

Note

Bibliography

**3. Probability and Randomness**

3.1 Introduction: probability, coin tossing and randomness

3.2 Expansions to a base

3.3 Rational numbers and periodic expansions

3.4 Sets, events, length and probability

3.5 Sets of measure zero

3.6 Independent sets and events

3.7 Typewriters, recurrence, and the Prince of Denmark

3.8 The Rademacher functions

3.9 Randomness, binary expansions and a law of averages

3.10 The dynamical systems approach

3.11 The Walsh functions

3.12 Normal numbers and randomness

3.13 Notions of probability and randomness

3.14 The curious phenomenon of the leading significant digit

3.15 Leading digits and geometric sequences

3.16 Multiple digits and a result of Diaconis

3.17 Dynamical systems and changes of scale

3.18 The equivalence of Kronecker and Benford dynamical systems

3.19 Scale invariance and the necessity of Benford's Law

Exercises

Investigations

Notes

Bibliography

**4. Recurrence**

4.1 Introduction: random systems and recurrence

4.2 Transformations that preserve length

4.3 Poincaré recurrence

4.4 Recurrent points

4.5 Kac's result on average recurrence times

4.6 Applications to the Kronecker and Borel dynamical systems

4.7 The standard deviation of recurrence times

Exercises

Investigations

Notes

Bibliography

**5. Averaging in Time and Space**

5.1 Introduction: averaging in time and space

5.2 Outer measure

5.3 Invariant sets

5.4 Measurable sets

5.5 Measure-preserving transformations

5.6 Poincaré recurrence...again!

5.7 Ergodic systems

5.8 Birkhoff's theorem: the time average equals the space average

5.9 Weyl's theorem from the ergodic viewpoint

5.10 The Ergodic Theorem and expansions to an arbitrary base

5.11 Kac's recurrence formula: the general case

5.12 Mixing transformations and an example of Kakutani

5.13 Lüroth transformations and continued fractions

Exercises

Investigations

Notes

Bibliography

**Index of Subjects**

**Index of Symbols**

**About the Author**

### 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).