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Invariant energy-weighted and regular-shell probabilities

Statement

Let $(M, ω, H, Φ, m_{0})$ be a Hamiltonian phase datum and let $R$ be a complete invariant Borel domain. If $w : R \to [0, \infty]$ is Borel and $0 < Z_{w} : = \int_{R} w (H) d m_{0} < \infty,$ then $μ_{w} (B) = Z_{w}^{- 1} \int_{B} w (H) d m_{0}$ is an invariant Borel probability on $R$ . In particular, for $β > 0$ , the canonical density $e^{- β H} / Z_{β}$ defines an invariant probability whenever $Z_{β} = \int_{R} e^{- β H} d m_{0}$ is finite and positive.

There is also the following regular-shell conclusion. Assume full AC and $n \geq 1$ . Let $O \subseteq R^{2 n}$ be open with canonical $ω = \sum_{i} d q^{i} \land d p_{i}$ , let $H \in C^{2} (O, R)$ , and let $J \subseteq R$ be open. Suppose $\nabla H \neq 0$ on $H^{- 1} (J)$ and $H^{- 1} (K)$ is compact in $O$ for every compact $K \subset J$ . For each energy $E \in J$ with nonempty shell $Σ_{E} = H^{- 1} ({E})$ , the weighted area measure $μ_{E}$ of Regular energy shell and weighted area measure satisfies $0 < μ_{E} (1) < \infty$ , and ${\hat{μ}}_{E} = μ_{E} / μ_{E} (1)$ is invariant under every Hamiltonian flow diffeomorphism preserving a full compact band containing $Σ_{E}$ . Critical levels and noncompact bands are outside this clause.

Facts & Assumptions

Given: The complete invariant domain and finite positive weighted normalizer in the first clause; the full-AC, regular-band hypotheses and band-preserving complete flow in the second clause.

[A1]

The shell clause assumes full AC as required by the inspected Coarea and invariant measure on compact regular energy shells and its The Axiom of Choice premise. This is stronger than the ${A C}_{ω}$ used to construct the Borel density measure.

[F1]

Hamiltonian phase, complete-domain flow and reference measure: $m_{0}$ is induced by $∣ ω^{n} / n! ∣ / a_{0}^{n}$ and $Φ_{t}$ is the actual Hamiltonian flow on $R$ .

[F2]

Liouville volume preservation: $Φ_{t}^{*} Ω = Ω$ on its actual domain for $Ω = ω^{n} / n!$ .

[F3]

A volume-preserving diffeomorphism preserves its induced Borel measure: a density-preserving self-diffeomorphism of a smooth manifold preserves its induced Borel measure. For the possibly different open flow domains, the required chartwise version is proved in step 1.1 before restricting to $R$ .

[F4]

A Hamiltonian is conserved along its own flow: $H \circ Φ_{t} = H$ wherever the autonomous Hamiltonian flow is defined.

[F5]

Integral invariance under measure-preserving maps: if $Φ_{t}$ preserves $m_{0}$ , then the integral of every nonnegative measurable function is unchanged by composition with $Φ_{t}$ .

[F6]

A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions: under countable choice, a $C^{1}$ diffeomorphism between open Euclidean sets transforms nonnegative Lebesgue integrals by its absolute Jacobian.

[F7]

Coarea and invariant measure on compact regular energy shells: for the exact hypotheses in the shell clause, its complete proof establishes the coarea identity, positive finite mass on every nonempty regular shell, and invariance of its normalized measure on the specified complete band. This current carrier is a published mathematical item; its statement and full proof were inspected, including its full-AC hypothesis and excluded critical/noncompact cases.

[F8]

Intrinsic density measure and its chart restriction: under countable choice, the measure of a Borel subset of a chart is the Lebesgue integral of the density coefficient in that chart.

Proof

Given: [A1] and [F1]–[F8].

1.1F1F2F3F6F8givenconstruct

Fix $t \in R$ and put $U_{t} = {z : (t, z) \in D}$ , where $D$ is the open flow domain of [F1]. The flow law makes $Φ_{t} : U_{t} \to U_{- t}$ a smooth diffeomorphism with inverse $Φ_{- t}$ . By [F2] it preserves the density $η_{0} = ∣ ω^{n} / n! ∣ / a_{0}^{n}$ . To extend the chart argument underlying [F3] to these two open sets, take countable chart covers and disjoint Borel refinements $(A_{i})$ of $U_{t}$ and $(B_{j})$ of $U_{- t}$ . For Borel $B \subseteq U_{- t}$ , put $S_{i j} = A_{i} \cap Φ_{t}^{- 1} (B_{j} \cap B)$ . In the corresponding charts the density coefficients obey $r_{x} = (r_{y} \circ T) ∣ \det D T ∣$ , with $T = y \circ Φ_{t} \circ x^{- 1}$ a diffeomorphism between open chart intersections. By [F6] and [F8], $m_{0} (S_{i j}) = m_{0} (Φ_{t} (S_{i j}))$ . Summing these nonnegative terms gives $m_{0} (Φ_{t}^{- 1} B) = m_{0} (B)$ . For $B \subseteq R$ , complete invariance implies $Φ_{t}^{- 1} B \subseteq R$ ; hence $Φ_{t} ∣_{R}$ preserves $m_{0} ∣_{R}$ , although $R$ need not be a manifold. Only countable choice from [F1] is needed here.

2.1F1F4F5step 1.1givenalgebra

By step 1.1, $Φ_{t} ∣_{R}$ preserves $m_{0} ∣_{R}$ , and [F4] gives $w (H (Φ_{t} z)) = w (H (z))$ for every $z \in R$ . Thus for any Borel $B \subseteq R$ , $μ_{w} (Φ_{t}^{- 1} B) = Z_{w}^{- 1} \int_{R} 1_{B} (Φ_{t} z) w (H (z)) d m_{0} (z) = Z_{w}^{- 1} \int_{R} (1_{B} w (H)) (Φ_{t} z) d m_{0} (z) = μ_{w} (B) .$ The last equality is [F5]. The finite positive normalizer makes $μ_{w}$ a probability, and the same calculation applies to $w (E) = e^{- β E}$ when $Z_{β} \in (0, \infty)$ .

3.1A1F6F7given∎

Under the shell clause's exact full-AC, $C^{2}$ , nonzero-gradient, compact-band hypotheses, [F6] is the chart substitution used in the coarea proof of [F7]. Apply [F7] to each $E \in J$ with nonempty shell and its specified complete band flow. It gives $0 < μ_{E} (1) < \infty$ and $μ_{E} (ϕ_{t}^{- 1} B) = μ_{E} (B)$ for every Borel shell set $B$ . Dividing by the same finite positive total mass proves invariance of ${\hat{μ}}_{E}$ . No conclusion is drawn for a critical shell or a noncompact band.

Remarks

The two clauses have different normalizability requirements: the energy weight must have a finite positive partition integral, while the shell clause requires the exact compact regular-band hypotheses above. Neither follows from Hamiltonian volume preservation alone.

Invariant energy-weighted and regular-shell probabilities

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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