False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)

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Weak mixing implies strong mixing

Statement

False claim: Every weakly mixing probability-preserving transformation is strongly mixing.

Facts & Assumptions

[F1]

Under AC, normalized Chacon is an invertible completed Lebesgue probability system that is weakly mixing but not strongly mixing Chacon transformation is weakly mixing but not mixing.

[F2]

Weak mixing uses absolute-Cesaro correlations, whereas strong mixing requires pointwise convergence in time for each fixed measurable pair Strong and weak mixing on a probability space.

[F3]

Assume AC The Axiom of Choice.

[F4]

The fixed Chacon set $A = [0, 2 / 9)$ satisfies $d_{h_{r}} (A, A) \geq 2 / 81$ for every $r \geq 1$ Chacon tower height correlations obstruct mixing, and $h_{r} = (3^{r + 1} - 1) / 2 \to \infty$ Chacon three cut one spacer towers.

Refutation

Given: AC.

1.1

Use the single probability system of F1. Its weak-mixing conclusion is precisely the premise in F2.

F1F2F3

2.1

In the system of step 1.1 the fixed pair $A = B = [0, 2 / 9)$ satisfies $d_{h_{r}} (A, A) \geq 2 / 81 > 0$ for all $r \geq 1$ by F4. Since $h_{r} \to \infty$ , these correlations do not converge to zero. Thus the conclusion of the false claim fails while its hypothesis holds. The example assumes AC exactly as F1 does; it does not infer pointwise convergence from an average.

F1F2F3F4step 1.1∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Katok–Thouvenot Theorem 5.12 p.697 (standard reference, not scraped)