Coarea and invariant measure on compact regular energy shells

Let $C$ be a compact energy band containing the support of $f$. At each $x\in H^{-1}(C)$, some coordinate derivative ${\mathrm{\partial }}_{i}H(x)$ is nonzero. By [F2] a neighborhood is parameterized by the remaining coordinates $z$ and energy $E$, with $x_i=h(z,E)$. Compactness gives a finite cover by such charts. Refine the cover by finitely many balls whose closures lie in chart domains and whose smaller concentric balls cover $H^{-1}(C)$. Choose standard smooth bump functions ${\psi }_a$ supported in the larger balls and positive on the smaller balls, and let $\mathrm{\Psi }={\sum }_a{\psi }_a$. Its minimum on $H^{-1}(C)$ is positive. Choose a smooth scalar cutoff $\eta$ that is zero below one quarter of this minimum and one above one half of it. Then ${\chi }_a=\eta (\mathrm{\Psi }){\psi }_a\mathrm{/}\mathrm{\Psi }$, extended by zero where $\mathrm{\Psi }=0$, is smooth, supported in its chart, and the sum is one near $H^{-1}(C)$.

At the stated $C^2$ regularity, $X_H$ in [F5] has continuous first derivatives. Along every flow curve, $\frac{d}{ds}H({\varphi }_s(x))={\sum }_j(H_{q_j}{H}_{p_j}-H_{p_j}{H}_{q_j})({\varphi }_s(x))=0$. Differentiating its integral equation ${\varphi }_s(x)=x+{\int }_0^s{X}_H({\varphi }_u(x))du$ in $x$ gives $J_s=I+{\int }_0^{s}D{X}_H({\varphi }_u(x))J_{u}du$, where $J_s=D_x{\varphi }_s(x)$; on each compact flow segment the $C^1$ derivatives are continuous, so difference quotients pass through this finite integral. Thus ${\mathrm{\partial }}_s{J}_s=D{X}_H({\varphi }_s(x))J_s$. Differentiating the determinant by multilinearity in its columns gives ${\mathrm{\partial }}_s\det J_s=(\mathrm{tr}D{X}_H)({\varphi }_s(x))\det J_s$. Indeed $J_s$ is invertible by the inverse flow, and the sum of determinants with one column replaced by $D{X}_H$ times that column is $\det J_s\mathrm{tr}(J_s^{-1}D{X}_H{J}_s)=\det J_s\mathrm{tr}D{X}_H$. Equality of the continuous mixed second derivatives of $H$ gives $\mathrm{tr}D{X}_H={\sum }_j({\mathrm{\partial }}_{q_j}{\mathrm{\partial }}_{p_j}H-{\mathrm{\partial }}_{p_j}{\mathrm{\partial }}_{q_j}H)=0$. Since $\det J_0=1$, $\det J_s=1$ throughout the flow domain; [F3] then gives Lebesgue-volume preservation. Both invariances therefore hold for the entire $C^2$ scope of this item.

In each chart, the map $(z,E)\mapsto (z,h_a(z,E))$ is injective after shrinking the chart, and its absolute Jacobian is $\mathrm{\mid }{\mathrm{\partial }}_E{h}_a\mathrm{\mid }=\mathrm{\mid }{\mathrm{\partial }}_{i}H{\mathrm{\mid }}^{-1}$. By [F3] and Tonelli [F4], ${\int }_{O}f(x){\chi }_a(x)dx={\int }_J\int f(z,h_a(z,E)){\chi }_a(z,h_a(z,E))\mathrm{\mid }{\mathrm{\partial }}_{i}H{\mathrm{\mid }}^{-1}dzdE\mathrm{.}$ Implicit differentiation gives ${\mathrm{\partial }}_{z_j}{h}_a=-{\mathrm{\partial }}_{j}H\mathrm{/}{\mathrm{\partial }}_{i}H$, so the graph area density is $\sqrt{1+\mathrm{\mid }{\mathrm{\nabla }}_z{h}_a{\mathrm{\mid }}^2}dz=\frac{\mathrm{\mid }\mathrm{\nabla }H\mathrm{\mid }}{\mathrm{\mid }{\mathrm{\partial }}_{i}H\mathrm{\mid }}dz\mathrm{.}$ The inner integral is therefore the corresponding chart contribution to the candidate ${\mu }_E(f)$ in [F1]. Sum the finitely many charts and use ${\sum }_a{\chi }_a=1$ on the support to obtain the asserted coarea identity.

On overlaps, if one graph parameterization is obtained from another by a $C^1$ coordinate change $\psi$, its Gram matrix is transformed by $G\mapsto (D\psi {)}^{\mathrm{\top }}G(D\psi )$. Taking determinants shows the area density changes by $\mathrm{\mid }\det D\psi \mathrm{\mid }$, exactly the ordinary chart Jacobian. Implicit differentiation gives $\sqrt{1+\mathrm{\mid }\mathrm{\nabla }h{\mathrm{\mid }}^2}\mathrm{/}\mathrm{\mid }\mathrm{\nabla }H\mathrm{\mid }=\mathrm{\mid }{\mathrm{\partial }}_{i}H{\mathrm{\mid }}^{-1}$, so the chart contributions define the same measure independent of the partition and equal ${\int }_{{\mathrm{\Sigma }}_E}g\mathrm{\mid }\mathrm{\nabla }H{\mathrm{\mid }}^{-1}d{A}_E$. Each shell is a closed subset of compact $H^{-1}(\{E\})$, hence compact. A finite collection of graph charts with $\mathrm{\nabla }H$ bounded away from zero on that compact level gives finite area and hence ${\mu }_E(1)<\mathrm{\infty }$. If the shell is nonempty, one graph chart contains an open patch; its coordinate domain has positive $(2n-1)$-dimensional measure and the continuous density $\mathrm{\mid }\mathrm{\nabla }H{\mathrm{\mid }}^{-1}d{A}_E\mathrm{/}dz$ is positive there, so ${\mu }_E(1)>0$.

Fix a compact band and a compactly supported continuous test $g$ on it. For a continuous compactly supported scalar function $q$ on $J$, the coarea identity from step 2.1 and the Hamiltonian volume/energy invariance from step 1.2 give ${\int }_{J}q(E){\mu }_E(g\circ {\varphi }_t)dE={\int }_{O}q(H(x))g({\varphi }_{t}x)dx={\int }_{O}q(H(x))g(x)dx={\int }_{J}q(E){\mu }_E(g)dE\mathrm{.}$ The second equality is the change of variables by the volume-preserving diffeomorphism ${\varphi }_t$; $H\circ {\varphi }_t=H$ keeps the energy weight fixed. For each test $g$, both functions of $E$ in the outer integrals are continuous locally: use the finite chart representation in step 2.1, compact chart support, the positive lower bound for $\mathrm{\mid }\mathrm{\nabla }H\mathrm{\mid }$, and dominated convergence [F6]. Their difference integrates to zero against every compactly supported continuous $q$. If the continuous difference were nonzero at one energy, it would keep one strict sign on a neighborhood; a nonnegative continuous bump $q$ supported there would make the integral nonzero. Thus ${\mu }_E(g\circ {\varphi }_t)={\mu }_E(g)$ for every $E$ in the band.

Fix a closed subset $F$ of a nonempty shell. If $F=\mathrm{\varnothing }$, both measures assign it zero. Otherwise let $\chi$ be a continuous compactly supported cutoff equal to one on the compact shell; the same finite-ball bump construction used in 1.1 supplies it. In the ambient band set $g_n(x)=\chi (x)\max \{0,1-n\mathrm{dist}(x,F)\}$. By [F7], $g_n$ is continuous, $0\le g_n\le 1$, and its restriction to ${\mathrm{\Sigma }}_E$ converges pointwise to ${\mathbf{1}}_F$. Because ${\varphi }_t$ maps the shell onto itself, $g_n\circ {\varphi }_t$ converges there to ${\mathbf{1}}_{{\varphi }_t^{-1}F}$. Apply step 3.1 to each $g_n$ and use dominated convergence [F6] for the finite shell measure from step 2.2; this gives ${\mu }_E({\varphi }_t^{-1}F)={\mu }_E(F)$ for every closed $F\subseteq {\mathrm{\Sigma }}_E$. Closed sets form a pi-system generating the shell Borel sigma-algebra, and both ${\mu }_E$ and its pushforward have the same finite total mass because ${\varphi }_t$ maps the shell onto itself.

By [F8], agreement on closed sets and equality of total masses imply that ${\mu }_E$ equals its pushforward on every Borel subset of ${\mathrm{\Sigma }}_E$. Divide by $0<{\mu }_E(1)<\mathrm{\infty }$ from step 2.2. Hence the normalized regular-shell probability is invariant for each such $E$, not merely almost every energy.