CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Weak mixing need not imply strong mixing

Statement refuted

Weak mixing implies strong mixing for every probability-preserving transformation.

Facts & Assumptions

Given: A measure-preserving probability system, the local definitions in Ergodicity and mixing for a selected probability map, and full AC.

[A1]

Full AC is assumed by the normalized Chacon construction and its published mathematical supplier (The Axiom of Choice).

[F1]

Ergodicity and mixing for a selected probability map: weak mixing uses $N^{- 1} \sum_{n < N} ∣ d_{n} (E, F) ∣ \to 0$ , while strong mixing requires $d_{n} (E, F) \to 0$ for each fixed measurable pair.

[F2]

Chacon transformation is weakly mixing but not mixing: under full AC the normalized three-cut one-spacer Chacon map on completed Lebesgue probability is weakly mixing in the absolute-Cesàro sense but is not strongly mixing. Its proof supplies the weak-mixing argument.

[F3]

Chacon tower height correlations obstruct mixing: for the fixed set $E = [0, 2 / 9)$ and unbounded tower heights $h_{r}$ , $μ (E \cap T^{- h_{r}} E) - μ (E)^{2} \geq 2 / 81 > 0$ .

Counterexample

Given: [A1], [F1]–[F3].

1.1A1F1F2construct

Choose the normalized three-cut one-spacer Chacon transformation supplied by [F2], on its completed Lebesgue probability space. The exact convention in [F1] matches the supplier's absolute-Cesàro definition, so this system is weakly mixing.

2.1A1F1F3step 1.1algebra∎

By [F3], the correlations for the fixed pair $(E, E)$ stay at least $2 / 81$ above zero along $h_{r} \to \infty$ . Strong mixing in [F1] would force these same correlations to tend to zero, which they do not. Hence this probability-preserving system is not strongly mixing and refutes the statement.

Remarks

This is a discrete measure-preserving example from ergodic theory. No Hamiltonian realization is claimed.

Weak mixing need not imply strong mixing

Statement refuted

Facts & Assumptions

Counterexample

Remarks

Depends on

Dependency tree · 2 levels

Sources