PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

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Integer-base circle maps preserve Lebesgue measure

Statement

Assume countable choice. For every integer $b \geq 2$ , the circle map $D_{b}$ is continuous, surjective and non-injective, and preserves Borel Lebesgue probability and its completion.

Facts & Assumptions

[F1]

The b affine branches and Lipschitz bound are explicit. Integer-base maps and b-adic circle intervals.

[F2]

The finite measure generator test applies with the whole space included. Measure preservation can be checked on a generating pi-system.

[F3]

Countable choice gives preservation on the completion. Compositions, iterates and completions preserve invariance.

[F4]

The nonzero dilation 1/b scales interval length by 1/b. For a nonzero real $c$ , dilation by $c$ multiplies Lebesgue outer measure by $∣ c ∣^{n}$ , and reflection in the origin preserves it.

Proof

Given: Assume countable choice. For every integer $b \geq 2$ , the circle map $D_{b}$ is continuous, surjective and non-injective, and preserves Borel Lebesgue probability and its completion.

1.1

For $0 \leq a < c \leq 1$ , $D_{b}^{- 1} [a, c) = ⋃_{j = 0}^{b - 1} [(a + j) / b, (c + j) / b)$ , with disjoint pieces of length $(c - a) / b$ . Thus their total measure is c-a. The empty interval has empty inverse image. The map is Borel measurable by the Lipschitz bound from its definition.

F1F4

2.1

The half-open intervals together with the empty set form a pi-system containing [0,1) and generating the circle Borel sets, as in the circle definition. Its whole-space mass is one, so the finite generator theorem gives Borel preservation; the completion theorem gives completed preservation. Countable choice is inherited by the Lebesgue, dilation and completion suppliers and is assumed here.

step 1.1F1F2F3

3.1

For every y in [0,1), the explicit preimage y/b lies in [0,1) and maps to y, proving surjectivity. The distinct points 0 and 1/b both map to 0, proving non-injectivity. Continuity is the already established inequality $d (D_{b} x, D_{b} y) \leq b d (x, y)$ .

step 2.1F1∎

Depends on

Used by

Base-three cylinders and their preimages Example
Doubling preserves Lebesgue measure Proposition
Every integer-base circle map is strongly mixing Theorem

Dependency tree · two levels

32 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

E–W Example 2.4 pp.14–15, b-branch generalization (standard reference, not scraped)