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

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Poincare recurrence for finite measure-preserving systems

Statement

In a measure-preserving system $(X, A, μ, T)$ with $μ (X) < \infty$ , for every $E \in A$ almost every $x \in E$ has $T^{n} x \in E$ for infinitely many positive integers $n$ .

Facts & Assumptions

[F1]

The set with no positive return and every one of its inverse-image levels are measurable and null. No-return sets have disjoint null preimage towers.

[F2]

A countable union of measurable null sets is null. Finite and countable subadditivity of measures.

Proof

Given: In a measure-preserving system $(X, A, μ, T)$ with $μ (X) < \infty$ , for every $E \in A$ almost every $x \in E$ has $T^{n} x \in E$ for infinitely many positive integers $n$ .

1.1

Put $W = E ∖ ⋃_{n \geq 1} T^{- n} E$ and $N = ⋃_{m \geq 0} T^{- m} W$ . Every set in this union is measurable and null, whence $N$ is measurable and $μ (N) = 0$ .

F1F2

2.1

If $x \in E$ has only finitely many positive return times, the finite nonempty set ${m \geq 0 : T^{m} x \in E}$ has a largest member $q$ (it contains zero). There is no positive return to $E$ from $T^{q} x$ , so $T^{q} x \in W$ and $x \in N$ . Thus every $x \in E ∖ N$ returns infinitely often. Conversely a point of $E \cap T^{- q} W$ has no visit after $q$ , so the exceptional set is exactly $E \cap N$ and is measurable. The argument includes the case of no positive return by taking $q = 0$ .

step 1.1given∎

Depends on

Used by

Topological recurrence on second-countable spaces Corollary
First-return times and induced transformations Definition
Recurrence to a dyadic interval under doubling Example
Induced transformations preserve restricted finite measure Theorem

Dependency tree · two levels

11 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 Theorem 2.11 p.21; Sarig Theorem 1.1 (standard reference, not scraped)