Contents

Mathematical symbols

1 Number-theoretical dynamical systems

1.1 Continued fractions and Diophantine approximation

1.1.1 Continued fractions

1.1.2 Elementary Diophantine approximation: Hurwitz’s Theorem and badly approximable numbers

1.2 Topological Dynamical Systems

1.2.1 The Gauss map

1.2.2 Symbolic dynamics

1.2.3 A return to the Gauss map

1.2.4 Elementary metrical Diophantine analysis

1.2.5 Markov partitions for interval maps

1.3 The Farey map: definition and topological properties

1.3.1 The Farey map

1.3.2 Topological properties of the Farey map

1.4 Two further examples

1.4.1 The a-Lüroth maps

1.4.2 The a-Farey maps

1.4.3 Topological properties of Fa

1.4.4 Expanding and expansive partitions

1.4.5 Metrical Diophantine-like results for the a-Lüroth expansion

1.5 Notes and historical remarks

1.5.1 The Farey sequence

1.5.2 The classical Lüroth series

1.6 Exercises

2 Basic ergodic theory

2.1 Invariant measures

2.1.1 Invariant measures for the Gauss and a-Lüroth system

2.2 Recurrence and conservativity

2.3 The transfer operator

2.3.1 Jacobians and the change of variable formula

2.3.2 Obtaining invariant measures via the transfer operator

2.3.3 The Ruelle operator

2.3.4 Invariant measures for F , Fa, G and La

2.3.5 Invariant measures via the jump transformation

2.4 Ergodicity and exactness

2.4.1 Ergodicity of the systems G and La

2.4.2 Ergodic theorems for probability spaces and consequences for the Gauss and a-Lüroth systems

2.4.3 Ergodic theorems for infinite measures

2.4.4 Inducing

2.4.5 Uniqueness of the invariant measures for F and Fa

2.4.6 Proof of Hopf’s Ratio Ergodic Theorem

2.5 Exactness revisited

2.6 Exercises

3 Renewal theory and a-sum-level sets

3.1 Sum-level sets

3.2 Sum-level sets for the a-Lüroth expansion

3.2.1 Classical renewal results

3.2.2 Renewal theory applied to the a-sum-level sets

3.3 Exercises

4 Infinite ergodic theory

4.1 The functional analytic perspective and the Chacon–Ornstein Ergodic Theorem

4.2 Pointwise dual ergodicity

4.3 ψ-mixing, Darling–Kac sets and pointwise dual ergodicity

4.4 Exercises

5 Applications of infinite ergodic theory

5.1 Sum-level sets for the continued fraction expansion, first investigations

5.2 ψ-mixing for the Gauss map and the Gauss problem

5.3 Pointwise dual ergodicity for the Farey map

5.4 Uniform and uniformly returning sets

5.5 Finer asymptotics of Lebesgue measure of sum-level sets

5.6 Uniform distribution of the even Stern–Brocot sequence

5.7 Exercises

Bibliography

Index