ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedaudited 2026-09-14

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A rational half-rotation has a nonconstant ergodic limit

Example

Assume the Axiom of Countable Choice. For $T = R_{1 / 2}$ on the circle and $f = 1_{[0, 1 / 4)}$ , the averages converge at every point to

$g = \frac{1}{2} 1_{[0, 1 / 4) \cup [1 / 2, 3 / 4)},$

a nonconstant invariant function.

Facts & Assumptions

Given: Countable choice, Lebesgue probability on the circle, $T = R_{1 / 2}$ , and $f = 1_{[0, 1 / 4)}$ .

[F1]

The half-rotation preserves Lebesgue probability (Circle rotations preserve Lebesgue measure).

[F2]

The average is $A_{n} f = n^{- 1} \sum_{k < n} f \circ T^{k}$ (Ergodic partial sums, time averages, and the invariant L2 subspace).

[F3]

Half-open intervals have Lebesgue measure equal to their length (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included).

[F4]

Countable choice is the standing assumption required by the circle-measure and interval-measure suppliers (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Verification

technique · direct period-two calculation

1.1

Since $T^{2}$ is the identity, the summands alternate between $f$ and $f \circ T$ . Moreover $f \circ T = 1_{[1 / 2, 3 / 4)}$ .

givenalgebra

2.1

For $n = 2 q$ , $A_{n} f = (f + f \circ T) / 2 = g$ . For $n = 2 q + 1$ , the counts of the two summands differ by one, so $∣ A_{n} f - g ∣ \leq 1 / n$ pointwise.

F2step 1.1

3.1

It follows that $A_{n} f \to g$ at every point. The half-rotation interchanges the two support intervals, so $g \circ T = g$ ; by [F3] it takes both values $0$ and $1 / 2$ on sets of positive measure and is therefore nonconstant.

F1F3step 1.1step 2.1

4.1

Finally, [F3] gives $\int g d λ = (1 / 2) (1 / 4 + 1 / 4) = 1 / 4 = \int f d λ$ . Thus the limit preserves the mean but need not equal the constant mean in this nonergodic example. Countable choice is used only through [F1] and [F3].

F1F3F4step 3.1algebra∎

Depends on

Used by

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Sources

Charles Walkden, Ergodic Theory lecture notes (standard reference, not scraped)