TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)

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Chacon transformation is weakly mixing but not mixing

Statement

Assume AC. There exists an invertible completed Lebesgue probability system, the normalized three-cut one-spacer Chacon transformation, that is weakly mixing in the absolute-Cesaro sense but is not strongly mixing.

Facts & Assumptions

[F1]

The Chacon partial maps extend to an invertible completed Lebesgue probability system Chacon partial maps extend to an invertible map mod null sets.

[F2]

Its complex $L^{2}$ eigenfunctions are constant Chacon eigenfunctions are constant.

[F3]

On such a probability space absence of nonconstant complex $L^{2}$ eigenfunctions is equivalent to absolute-Cesaro weak mixing Weak mixing is equivalent to absence of nonconstant eigenfunctions.

[F4]

The fixed set $A = [0, 2 / 9)$ has correlations along $h_{r}$ at least $2 / 27$ , exceeding $μ (A)^{2} = 4 / 81$ Chacon tower height correlations obstruct mixing.

[F5]

Assume AC The Axiom of Choice.

Proof

Given: AC and the normalized Chacon construction.

1.1

Take the ambient transformation furnished by F1, or its invariant conull restriction. It preserves completed Lebesgue probability and has a measurable inverse modulo null sets, meeting all F3 hypotheses. F2 excludes every nonconstant complex $L^{2}$ eigenfunction, so F3 gives weak mixing with the absolute value inside the Cesaro average.

F1F2F3F5

2.1

In the same system, F4 gives a fixed measurable $A$ of measure $2 / 9$ and unbounded heights $h_{r}$ with $μ (A \cap T^{- h_{r}} A) - μ (A)^{2} \geq 2 / 81 > 0$ . These correlations cannot tend to zero, so this system is not strongly mixing. The null-set modification of F1 leaves these measures unchanged. Together with step 1.1 this supplies the claimed witness, with AC inherited from every local construction and spectral input.

F4F5step 1.1∎

Depends on

Used by

Weak mixing implies strong mixing False statement

Dependency tree · two levels

25 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Peter Varju, Topics in Ergodic Theory, Michaelmas 2016, section 11 pp.36–40 (complete Chacon argument; public mirror) (standard reference, not scraped)
Katok–Thouvenot Theorem 5.12 pp.696–697 (standard reference, not scraped)
Sarig Problems 3.8–3.10 pp.99–101 (standard reference, not scraped)