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 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 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 times -- Applications 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 Kakutani -- Lüroth transformations and continued fractions -- Exercises -- Investigations -- Notes -- Bibliography -- Bibliography -- Index of Subjects -- Index of Symbols -- About the Author.