TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Hamiltonian recurrence under the exact finite-measure hypothesis

Statement

Let $(M, ω, H, Φ, m_{0})$ be a Hamiltonian phase datum, let $R \subseteq M$ be a complete invariant Borel domain, and fix $Δ > 0$ . Put $T = Φ_{Δ} ∣_{R}$ . Let $μ$ be any finite Borel measure on $R$ invariant under $T$ . Then for every Borel $E \subseteq R$ , $μ$ -almost every $z \in E$ satisfies $T^{k} z \in E$ for infinitely many integers $k \geq 1$ . The result is almost-everywhere recurrence for this fixed sampled map; it is not a pointwise statement about every initial state. The hypothesis also allows finite invariant measures singular with respect to ambient Liouville volume.

Facts & Assumptions

Given: A complete invariant Borel domain $R$ , a time step $Δ > 0$ , and a finite Borel measure $μ$ on $R$ invariant under $T = Φ_{Δ} ∣_{R}$ .

[F1]

Hamiltonian phase, complete-domain flow and reference measure: $Φ_{t}$ is the actual local Hamiltonian flow, and completeness and invariance on $R$ mean it is defined there for all real $t$ and maps $R$ onto itself.

[F2]

Measure-preserving transformations and systems: invariance is $μ (T^{- 1} B) = μ (B)$ for every Borel $B \subseteq R$ .

[F3]

Poincare recurrence for finite measure-preserving systems: for a measure-preserving self-map of a finite measure space, almost every point of each measurable set returns to it infinitely many positive iterates.

Proof

Given: The phase datum, $R$ , $Δ$ , and [F1]–[F3].

1.1F1F2given

By [F1], the flow is defined on every point of $R$ at time $Δ$ and maps $R$ onto itself, so $T$ is a measurable self-map. By [F2] and the hypothesis, $T$ preserves the finite measure $μ$ .

2.1F3step 1.1∎

Apply [F3] to $(R, B (R), μ, T)$ and the arbitrary Borel set $E$ . Its conclusion is exactly that $μ$ -almost every point of $E$ returns to $E$ infinitely many positive integer iterates.

Remarks

The theorem does not assume or prove finite total Liouville volume for $M$ . An infinite-volume phase space can still have a finite invariant probability or a finite invariant restriction, but that is a separate hypothesis.

Hamiltonian recurrence under the exact finite-measure hypothesis

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources