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Weak Mixing and the Chacon Transformation — Examples
1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Areas of Elementary Plane Figures
- Binary Operations, Monoids, Groups and Subgroups
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Complex Lp Spaces and Test-Function Conventions
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Countability Axioms and Cardinal Functions
- Density Separability and Convolution in Lᵖ
- Determinants of Matrices over a Commutative Ring
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Lebesgue Measure on Euclidean Space
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Measurable Functions and Simple Approximation
- Measure-Preserving Systems and Mixing Criteria
- Measures and Their Basic Properties
- Metric Spaces
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal Subgroups and Quotient Groups
- Order, Zorn's Lemma, and the Axiom of Choice
- Outer Measure and the Caratheodory Extension Theorem
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Product Measures and the Fubini Tonelli Theorems
- Properties of the Integral and the Working FTC
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Separation Axioms: the Hierarchy
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sigma Algebras and Borel Sets
- Simple Field Extensions and the Construction of the Complex Numbers
- Subspaces, Products, and Quotients
- Suprema and Infima
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Lebesgue Integral and the Convergence Theorems
- The Logarithm and General Powers
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Topology of Euclidean Space
- The Total Derivative in ℝᵐ → ℝⁿ
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
- Weak Mixing and the Chacon Transformation
2 · Summary
The tower recurrence gives the first three heights and the closed formula . A separate calculation tracks spacer widths, finite geometric sums and the reservoir tending to zero. These computations fix the indexing used in the construction.
The correlation example keeps one interval fixed while the return times vary. Its positive gap shows why the failure of strong mixing is a failure of a limit for a single pair of sets. It does not rely on selecting a different witness at every stage. The source obligations these examples inherit from the construction are the ones recorded as resolved for the companion page's batch coverage, not a fresh claim about those sources.
3 · Logical flowchart
4 · Definitions, theorems and proofs
None yet.
5 · Examples, counterexamples and false statements
First three chacon tower heights
Example
The first three formal Chacon tower heights are , , . For every integer , .
Facts & Assumptions
The finite tower recursion is , Chacon three cut one spacer towers. Its finite combinatorial clause is choice-free.
Verification
Given: The formal recursion of F1.
Direct substitution gives and . The extra one counts the spacer level, whereas the three copies count the old levels in the three columns.
Set . Then and . Induction on using the same initial value and recurrence proves . At this gives and , agreeing with step 1.1. Only finite arithmetic and induction are used; no measure existence or choice assumption is used.
Chacon spacer measure budget
Example
Assume AC for the Lebesgue-measure interpretation. The initial Chacon column has mass . The first two spacers have masses and . All spacers together have mass ; after stage the unused reservoir has mass . Spacers are retained, so it is the unused tail, not total spacer mass, that tends to zero.
Facts & Assumptions
Spacer has width , the initial column has width , and all the physical spacers are disjoint and retained Chacon three cut one spacer towers.
Assume AC for the measure assertions in F1 The Axiom of Choice.
Verification
Given: The finite normalized Chacon construction.
At the first cut the spacer has width , and at the second it has width . After stage the retained spacer mass is the finite sum . The empty sum at is zero. For , multiplying the sum by cancels all interior terms, giving , hence . The formula also gives zero at .
Adding the initial mass yields , whose missing mass is exactly the reservoir. As , and the reservoir tends to zero. Countable additivity on the disjoint spacer intervals identifies their union's measure with this sum; their union is also the physical interval , since their adjacent endpoints tend to one. The numerical geometric identities themselves are choice-free; AC is used only through F1's Lebesgue-measure interpretation.
Chacon correlation subsequence prevents mixing
Statement refuted
The normalized Chacon transformation is strongly mixing.
Facts & Assumptions
For Chacon, has measure and for all Chacon tower height correlations obstruct mixing. The heights satisfy Chacon three cut one spacer towers.
Assume AC, as in F1 The Axiom of Choice.
Counterexample
Given: The Chacon probability system under AC.
Use the fixed pair from F1. The product of its measures is , while its correlations at the stated heights are at least . Their difference is therefore at least for every .
A sequence tending to zero must eventually have absolute value below on every subsequence whose indices tend to infinity. The correlations minus their product violate this necessary condition along . Thus the strong-mixing conclusion refuted by F1 fails, without any need to assert that the correlations have a subsequential limit. The witness set is fixed, not chosen anew at each height.