Einsiedler M., Ward T. Ergodic Theory with a View to Number Theory

London: Springer, 2011. — 497 p.This text is a rigorous introduction to ergodic theory, developing the machinery of conditional measures and expectations, mixing, and recurrence.
Beginning by developing the basics of ergodic theory and progressing to describe some recent applications to number theory, this book goes beyond the standard texts in this topic. Applications include Weyl's polynomial equidistribution theorem, the ergodic proof of Szemeredi's theorem, the connection between the continued fraction map and the modular surface, and a proof of the equidistribution of horocycle orbits.
Ergodic Theory with a view towards Number Theory will appeal to mathematicians with some standard background in measure theory and functional analysis. No background in ergodic theory or Lie theory is assumed, and a number of exercises and hints to problems are included, making this the perfect companion for graduate students and researchers in ergodic theory, homogenous dynamics or number theory.Motivation
Examples of Ergodic Behavior
Equidistribution for Polynomials
Szemer ́edi’s Theorem
Indefinite Quadratic Forms and Oppenheim’s Conjecture
Littlewood’s Conjecture
Integral Quadratic Forms
Dynamics on Homogeneous Spaces
An Overview of Ergodic Theory
Ergodicity, Recurrence and Mixing
Measure-Preserving Transformations
Recurrence
Ergodicity
Associated Unitary Operators
The Mean Ergodic Theorem
Pointwise Ergodic Theorem
The Maximal Ergodic Theorem
Maximal Ergodic Theorem via Maximal Inequality
Maximal Ergodic Theorem via a Covering Lemma
The Pointwise Ergodic Theorem
Two Proofs of the Pointwise Ergodic Theorem
Strong-Mixing and Weak-Mixing
Proof of Weak-Mixing Equivalences
Continuous Spectrum and Weak-Mixing
Induced Transformations
Continued Fractions
Elementary Properties
The Continued Fraction Map and the Gauss Measure
Badly Approximable Numbers
Lagrange’s Theorem
Invertible Extension of the Continued Fraction Map
Invariant Measures for Continuous Maps
Existence of Invariant Measures
Ergodic Decomposition
Unique Ergodicity
Measure Rigidity and Equidistribution
Equidistribution on the Interval
Equidistribution and Generic Points
Equidistribution for Irrational Polynomials
Conditional Measures and Algebras
Conditional Expectation
Martingales
Conditional Measures
Algebras and Maps
Factors and Joinings
The Ergodic Theorem and Decomposition Revisited
Invariant Algebras and Factor Maps
The Set of Joinings
Kronecker Systems
Constructing Joinings
Furstenberg’s Proof of Szemeredi’s Theorem
Van der Waerden
Multiple Recurrence
Reduction to an Invertible System
Reduction to Borel Probability Spaces
Reduction to an Ergodic System
Furstenberg Correspondence Principle
An Instance of Polynomial Recurrence
The van der Corput Lemma
Two Special Cases of Multiple Recurrence
Kronecker Systems
Weak-Mixing Systems
Roth’s Theorem
Proof of Theorem 7.14 for a Kronecker System
Reducing the General Case to the Kronecker Factor
Definitions
Dichotomy Between Relatively Weak-Mixing and Compact Extensions
SZ for Compact Extensions
SZ for Compact Extensions via van der Waerden
A Second Proof
Chains of SZ Factors
SZ for Relatively Weak-Mixing Extensions
Concluding the Proof
Further Results in Ergodic Ramsey Theory
Other Furstenberg Ergodic Averages Actions of Locally Compact Groups
Actions of Locally Compact Groups
Ergodicity and Mixing
Mixing for Commuting Automorphisms
Ledrappier’s “Three Dots” Example
Mixing Properties of the ×2, ×3 System
Haar Measure and Regular Representation
Measure-Theoretic Transitivity and Uniqueness
Amenable Groups
Definition of Amenability and Existence of Invariant Measures
Mean Ergodic Theorem for Amenable Groups
Pointwise Ergodic Theorems and Polynomial Growth
Flows
Pointwise Ergodic Theorems for a Class of Groups
Ergodic Decomposition for Group Actions
Stationary Measures Geodesic Flow on Quotients of the Hyperbolic Plane
Geodesic Flow on Quotients of the Hyperbolic Plane
The Hyperbolic Plane and the Isometric Action
The Geodesic Flow and the Horocycle Flow
Closed Linear Groups and Left Invariant Riemannian Metric.
The Exponential Map and the Lie Algebra of a Closed Linear Group
The Left-Invariant Riemannian Metric
Discrete Subgroups of Closed Linear Groups
Dynamics on Quotients
Hyperbolic Area and Fuchsian Groups
Dynamics on Γ \ PSL 2 (R)
Lattices in Closed Linear Groups
Hopf’s Argument for Ergodicity of the Geodesic Flow
Ergodicity of the Gauss Map
Invariant Measures and the Structure of Orbits
Symbolic Coding
Measures Coming from Orbits
Nilrotation
Rotations on the Quotient of the Heisenberg Group
The Nilrotation
First Proof of Theorem 10.1
Second Proof of Theorem 10.1
A Commutative Lemma; The Set K
Studying Divergence; The Set X 1
Combining Linear Divergence and the Maximal Ergodic Theorem
A Non-ergodic Nilrotation
The General Nilrotation
More Dynamics on Quotients of the Hyperbolic Plane
Dirichlet Regions
Examples of Lattices
Arithmetic and Congruence Lattices in SL 2 (R)
A Concrete Principal Congruence Lattice of SL 2 (R)
Uniform Lattices
Unitary Representations, Mautner Phenomenon, and Ergodicity
Three Types of Actions
Ergodicity
Mautner Phenomenon for SL 2 (R)
Mixing and the Howe–Moore Theorem
First Proof of Theorem 11.22
Vanishing of Matrix Coefficients for PSL 2 (R)
Second Proof of Theorem 11.22; Mixing of All Orders.
Rigidity of Invariant Measures for the Horocycle Flow
Existence of Periodic Orbits; Geometric Characterization
Proof of Measure Rigidity for the Horocycle Flow
Non-escape of Mass for Horocycle Orbits
The Space of Lattices and the Proof of Theorem 11. for X 2 = SL 2 (Z)\ SL 2 (R)
Extension to the General Case
Equidistribution of Horocycle Orbits
Appendix A: Measure Theory
A.1 Measure Spaces
A.2 Product Spaces
A.3 Measurable Functions
A.4 Radon–Nikodym Derivatives
A.5 Convergence Theorems
A.6 Well-Behaved Measure Spaces
A.7 Lebesgue Density Theorem
A.8 Substitution Rule
Appendix B: Functional Analysis
B.1 Sequence Spaces
B.2 Linear Functionals
B.3 Linear Operators
B.4 Continuous Functions
B.5 Measures on Compact Metric Spaces
B.6 Measures on Other Spaces
B.7 Vector-valued Integration
Appendix C: Topological Groups
C.1 General Definitions
C.2 Haar Measure on Locally Compact Groups
C.3 Pontryagin Duality
Hints for Selected Exercises