Alphabeta MathMeasure Preserving Transformations and Poincare Recurrence
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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
- 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
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
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Sigma Algebras and Borel Sets
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Maximal Function and Lebesgue Differentiation
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Approximation and Compactness in C(K)
- 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
- 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
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- 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
Lemma
No-return sets have disjoint null preimage towers
Theorem
Poincare recurrence for finite measure-preserving systems
Corollary
Topological recurrence on second-countable spaces
Definition
First-return times and induced transformations
Proposition
First-return time and induced map are measurable
Theorem
Induced transformations preserve restricted finite measure
Proposition
Inducing an ergodic system gives an ergodic system
Theorem
Kac return-time formula without invertibility
Theorem
Kac integral formula for excursions
Definition
The circle, rotations and the doubling map
Proposition
Circle rotations preserve Lebesgue measure
Lemma
Irrational circle orbits are dense
Theorem
Circle rotation is ergodic for Lebesgue measure exactly at irrational angles
Definition
Integer-base maps and b-adic circle intervals
Proposition
Integer-base circle maps preserve Lebesgue measure
Theorem
Every integer-base circle map is strongly mixing
Proposition
Doubling preserves Lebesgue measure
Theorem
Doubling is ergodic for Lebesgue measure
Proposition
Doubling is strongly mixing for Lebesgue measure
Definition
Binary-sequence cylinders and fair-coin content
Lemma
Binary-sequence space is compact without Tychonoff
Lemma
Fair-coin cylinder content is a premeasure
Theorem
Fair-coin measure on binary sequences
Theorem
The fair-coin one-sided shift preserves measure and is mixing
Lemma
A countable dense family of continuous functions on a compact metric space
Lemma
A compact-metric probability representation using countable choice
Lemma
Probability sequences on compact metric spaces have integral-convergent subsequences
Theorem
Krylov–Bogolyubov existence of an invariant probability
False statement
False: measure-preserving transformations are invertible
False statement
False: ergodicity implies strong mixing
False statement
False: Poincare recurrence needs no finite total measure
False statement
False: an ergodic invariant sigma-algebra has only two sets
False statement
False: every orbit of an ergodic system is dense
False statement
False: constant continuous invariants characterize measure ergodicity
Sources
- E–W Theorem 2.11 p.21
- E–W Theorem 2.11 p.21; Sarig Theorem 1.1
- E–W Exercise 2.2.3 pp.22–23
- Sarig Definition 1.18 pp.28–29; domain correction
- Sarig §1.6.4 p.28
- Sarig Theorem 1.7(1), p.28
- Sarig Theorem 1.7(2), pp.28–29
- Sarig Theorem 1.7(3), pp.28–29 (specialization f=1)
- Sarig Theorem 1.7(3), pp.28–29
- E–W Examples 2.2 and 2.4 pp.14–15
- E–W Example 2.2 p.14
- E–W Proposition 2.16 p.26 and Example 2.33 p.49
- E–W Exercise 2.4.1 p.32; Proposition 2.16 conclusion
- E–W Example 2.4 pp.14–15; MT-22 binding base-b amendment
- E–W Example 2.4 pp.14–15, b-branch generalization
- Sarig Proposition 1.5 p.9, b-adic branch-count generalization
- E–W Example 2.4 pp.14–15
- Sarig Proposition 1.5 p.9, conclusion; density argument is the MT-22 prescribed local replacement
- Sarig Proposition 1.5 p.9
- E–W Examples 2.8–2.9 pp.17–18
- E–W Examples 2.8–2.9 compact metric footnote; elementary binary-tree proof
- E–W Examples 2.8–2.9; local compactness construction
- E–W Examples 2.8–2.9 with local Caratheodory route
- E–W Proposition 2.15; Exercise 2.7.9
- E–W §1.4 pp.97–98, replacing weak-star compactness with the design-required local argument
- Compact-metric adaptation of the RMK construction; Cohn, Measure Theory, 2nd ed., Chapter 7
- E–W §3.1 pp.97–98; local diagonal/RMK replacement for source weak-star compactness
- E–W Theorem 4.1 and Corollary 4.2 p.98
- E–W Example 2.4
- E–W Example 2.33
- E–W Example 2.12 pp.21–22
- E–W Proposition 2.14; doubling example specialization
- Sarig Proposition 1.5, fixed-point specialization
- E–W ergodicity definition and doubling example; explicit measure specialization