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Chacon tower height correlations obstruct mixing

Statement

Assume AC. For the fixed set $A = L_{1, 0} = [0, 2 / 9)$ and every $r \geq 1$ , $μ (A \cap T^{- h_{r}} A) \geq 2 / 27 > 4 / 81 = μ (A)^{2}$ . Consequently Chacon is not strongly mixing. More generally, every measurable $E \subseteq [0, 1)$ satisfies ${lim inf}_{r \to \infty} μ (E \cap T^{- h_{r}} E) \geq μ (E) / 3$ .

Facts & Assumptions

[F1]

The next Chacon tower stacks left thirds, middle thirds, one spacer and right thirds in that order, with $h_{r} = (3^{r + 1} - 1) / 2$ Chacon three cut one spacer towers.

[F2]

The limiting probability transformation agrees with finite tower arrows off a fixed null set Chacon partial maps extend to an invertible map mod null sets.

[F3]

Every measurable set has tower-level-union approximants with symmetric-difference error tending to zero Chacon levels approximate measurable sets.

[F4]

Strong mixing requires convergence of every fixed set-pair correlation to the product of the measures Strong and weak mixing on a probability space.

[F5]

Assume AC The Axiom of Choice.

Proof

Given: The normalized Chacon towers and their transformation under AC.

1.1

If $Q$ is any union of levels at stage $r$ , let $Q^{(0)}$ be the union of their left thirds. These disjoint thirds have total measure $μ (Q) / 3$ . F1's ordering and F2 show $T^{h_{r}} Q^{(0)}$ is the union of the corresponding middle thirds, modulo the fixed null set. Both unions lie in $Q$ , so $Q^{(0)} \subseteq Q \cap T^{- h_{r}} Q$ modulo null sets, and $μ (Q \cap T^{- h_{r}} Q) \geq μ (Q) / 3$ .

F1F2F5

2.1

The stage-one base is the left third of $[0, 2 / 3)$ , hence $A = [0, 2 / 9)$ with measure $2 / 9$ . At every later stage it is exactly the union of all its descendant levels, since each old level partitions into three retained thirds. Step 1.1 gives the bound $μ (A) / 3 = 2 / 27$ for every $r \geq 1$ . But $μ (A)^{2} = 4 / 81$ and $2 / 27 - 4 / 81 = 2 / 81 > 0$ . The heights $h_{r} \to \infty$ by F1. Thus this one fixed pair $(A, A)$ fails F4's limit, proving failure of strong mixing.

F1F4step 1.1

3.1

For measurable $E$ , F3 gives stage- $r$ level unions $Q_{r}$ with $η_{r} = μ (E △ Q_{r}) \to 0$ . The symmetric difference of $E \cap T^{- h_{r}} E$ and $Q_{r} \cap T^{- h_{r}} Q_{r}$ lies in $(E △ Q_{r}) \cup T^{- h_{r}} (E △ Q_{r})$ . Measure preservation bounds its measure by $2 η_{r}$ , while $μ (Q_{r}) \geq μ (E) - η_{r}$ . Step 1.1 therefore yields $μ (E \cap T^{- h_{r}} E) \geq μ (E) / 3 - (7 / 3) η_{r}$ . Taking the liminf proves the general assertion. AC is inherited from F1–F3 and permits the countable choice of approximants; the estimate remains valid for null or conull $E$ without division.

F2F3F5step 1.1∎

Depends on

Used by

Chacon correlation subsequence prevents mixing Counterexample
Weak mixing implies strong mixing False statement
Chacon transformation is weakly mixing but not mixing Theorem

Dependency tree · two levels

19 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 proof p.697 (standard reference, not scraped)
Sarig Problem 3.10 p.101 (standard reference, not scraped)
Creutz Theorem 6.11 p.42 (standard reference, not scraped)