Alphabeta MathWeak Mixing and the Chacon Transformation
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Measure Theory
- Complex Lp Spaces and Test-Function Conventions
- Density Separability and Convolution in Lᵖ
- Lebesgue Measure on Euclidean Space
- Measurable Functions and Simple Approximation
- Measure-Preserving Systems and Mixing Criteria
- Measures and Their Basic Properties
- Modes of Convergence Egorov and Lusin
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Product Measures and the Fubini Tonelli Theorems
- Sigma Algebras and Borel Sets
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Areas of Elementary Plane Figures
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Fundamental Trigonometric Identities
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Definition
Eigenfunction for a probability system
Definition
L two operator conventions for weak mixing
Lemma
Closed l two subspaces have orthogonal projections
Lemma
Hilbert cesaro averages converge to the fixed subspace
Lemma
Product rectangle kernels are dense in complex l two
Lemma
Square integrable kernels define bounded compact integral operators
Lemma
Conjugate transpose kernels give adjoints
Lemma
Invariant square integrable kernel produces a compact intertwiner
Lemma
Nonzero compact kernel operators yield nonzero positive compact k star k
Lemma
The positive norm eigenvalue of a nonzero positive compact self-adjoint operator has a nonzero finite-dimensional eigenspace
Lemma
Compact intertwiners produce finite dimensional invariant subspaces
Lemma
Nonzero finite dimensional complex invariant subspaces have unitary eigenvectors
Theorem
Weak mixing is equivalent to absence of nonconstant eigenfunctions
Definition
Chacon three cut one spacer towers
Lemma
Chacon partial maps extend to an invertible map mod null sets
Lemma
Chacon levels approximate measurable sets
Theorem
Chacon transformation is ergodic
Lemma
Chacon eigenfunctions are constant
Lemma
Chacon tower height correlations obstruct mixing
Theorem
Chacon transformation is weakly mixing but not mixing
False statement
Weak mixing implies strong mixing
Sources
- Sarig Definition 3.3 p.90; Definition 3.5 p.91
- Axler 8B, 10A, 10C definitions; Example 10.5
- Axler 8.28 p.226, 8.37–8.40 pp.228–229
- Sarig Theorem 2.1 pp.35–36
- Axler 8.37–8.40 projection route
- Axler 10.70 p.314; local generating-algebra proof replaces the last section-quantifier step
- Axler Example 10.5 p.282; 10.67–10.70 pp.312–314
- Axler Example 10.5 equations 10.6–10.9 p.282
- Sarig Theorem 3.2 p.91 (criterion); Axler Example 10.5 p.282 (kernel formula)
- Axler Example 10.5 and 10.70; intertwining derived locally
- Axler Example 10.5; 10.69(b) p.313 and 10.96 p.326; direct K-star-K argument
- Axler 10.96–10.99 p.326, with local positive quadratic argument replacing Fredholm theory
- Sarig Theorem 3.2 p.91, local compact-intertwiner proof of its difficult implication
- Axler 10.99 p.326 (local positive version)
- Sarig Definition 3.3 p.90; finite-dimensional eigenvalue interface proved by the published characteristic-polynomial and minimum-modulus suppliers
- Sarig Theorem 3.2 pp.91–92; local Hilbert and compact-kernel route supplies implication omitted there
- Axler Example 10.5, 10.70, 10.96–10.99 (local variants)
- Peter Varju, Topics in Ergodic Theory, Michaelmas 2016, section 11 pp.36–40 (complete Chacon argument; public mirror)
- Sarig Problem 3.8 pp.99–101
- Katok–Thouvenot §5.2.3 pp.696–697
- Sarig Problem 3.8(1–2) p.101
- Katok–Thouvenot construction pp.696–697
- Sarig Problem 3.8(3) p.101
- Katok–Thouvenot generating partitions paragraph p.697
- Katok–Thouvenot rank-one generating partitions paragraph p.697, expanded local ergodicity argument
- Sarig Problem 3.8–3.9 p.101
- Katok–Thouvenot Theorem 5.12 proof p.697
- Sarig Problem 3.9 p.101
- Creutz Theorem 6.11 and Exercise 6.3 pp.42–43 (incomplete source; local completion above)
- Sarig Problem 3.10 p.101
- Creutz Theorem 6.11 p.42
- Katok–Thouvenot Theorem 5.12 pp.696–697
- Sarig Problems 3.8–3.10 pp.99–101
- Katok–Thouvenot Theorem 5.12 p.697