Coarea and invariant probability on compact regular phase shells

In a canonical open phase domain $O\subseteq {\mathbb{R}}^{2n}$ with $m_N=c_{N}dz$, hypotheses [A1]–[A2] match the published Euclidean theorem [F2]. It yields the graph-partition measure, the coarea identity, and finite positive mass on each nonempty regular level. Multiplying both the volume measure and every shell measure by the same positive constant $c_N$ preserves the identity and leaves $P_E$ unchanged.

For a compact cotangent energy band, choose finitely many canonical charts whose smaller coordinate neighborhoods cover the band and whose closures lie inside the charts. In chart coordinates, $b(s)=e^{-1\mathrm{/}s}$ for $s>0$ and $b(s)=0$ for $s\le 0$ is smooth. For a coordinate ball of squared radius $R^2$ centered at $z_0$, $b(R^2-\mathrm{\mid }z-z_0{\mathrm{\mid }}^2)$ is a smooth bump, positive on the ball and zero outside its closure. Choose finitely many such balls whose smaller concentric balls cover the band and whose closures lie inside charts. Dividing the bumps by their positive sum near the band gives a finite partition ${\sum }_a{\chi }_a=1$ on a neighborhood of the band. This uses only a finite subcover. Refine it so that one fixed ${\mathrm{\partial }}_{i}H$ is nonzero on each chart support. By [F3], each local level is a graph $z_i=h_a(y,E)$. The map $(y,E)\mapsto (y,h_a(y,E))$ has absolute Jacobian $\mathrm{\mid }{\mathrm{\partial }}_E{h}_a\mathrm{\mid }=\mathrm{\mid }{\mathrm{\partial }}_{i}H{\mathrm{\mid }}^{-1}$, so [F4] and [F5] give the coarea identity for $f{\chi }_a$. Implicit differentiation gives ${\mathrm{\partial }}_y{h}_a=-{\mathrm{\partial }}_{y}H\mathrm{/}{\mathrm{\partial }}_{i}H$, hence $\sqrt{1+\mathrm{\mid }{\mathrm{\nabla }}_y{h}_a{\mathrm{\mid }}^2}=\mathrm{\mid }{\mathrm{\nabla }}_{z}H\mathrm{\mid }\mathrm{/}\mathrm{\mid }{\mathrm{\partial }}_{i}H\mathrm{\mid }\mathrm{.}$ Thus the chart term is exactly the displayed weighted area integral. Cotangent coordinate changes have block Jacobian $\left(\begin{array}{cc}Dq& 0\\ \ast & D{q}^{-\mathsf{T}}\end{array}\right),$ whose determinant has absolute value one; therefore the Liouville density and the chart integrals agree on overlaps. Summing the finite partition proves the coarea identity and partition independence on the cotangent band.

The shell is a closed subset of a compact inverse band, hence compact. A finite set of graph charts with $\mathrm{\mid }{\mathrm{\nabla }}_{z}H\mathrm{\mid }$ bounded away from zero gives ${\mu }_E(1)<\mathrm{\infty }$. A nonempty shell contains a graph patch with a nonempty open coordinate domain and positive continuous chart density, so ${\mu }_E(1)>0$.

Along a Hamiltonian trajectory, $\frac{d}{dt}H({\mathrm{\Phi }}_{t}z)=dH(X_H)({\mathrm{\Phi }}_{t}z)={\omega }_N(X_H,X_H)=0$; therefore the flow preserves energy. The $C^2$ determinant argument in [F7] gives absolute coordinate Jacobian one for each canonical local flow. Since the reference density is $c_{N}dz$, [F4] proves measure preservation on each chart; composing local flow maps retains this property. Refine the finite source and target chart covers of the compact band into disjoint Borel pieces on which each chart representation of ${\mathrm{\Phi }}_t$ applies, and sum the COV identities to obtain preservation of $m_N$ on the band. Write the fixed band as $D=H^{-1}([a,b])$, with $[a,b]\subset J$ and the shell of interest at an energy in $(a,b)$. For continuous $f$ on $D$ and $q\in C_c((a,b))$, all following compositions are evaluated only on $D$. By [F7]'s integral-invariance clause and step 2.1, ${\int }_a^{b}q(E){\mu }_E(f\circ {\mathrm{\Phi }}_t)dE={\int }_{D}q(H(z))f({\mathrm{\Phi }}_{t}z)d{m}_N={\int }_{D}q(H(z))f(z)d{m}_N={\int }_a^{b}q(E){\mu }_E(f)dE\mathrm{.}$ The coarea applications extend these products by zero outside $D$; the support of $q$ lies away from the band endpoints, so the extensions have compact support in $H^{-1}(J)$.

On any compact subinterval of $(a,b)$, use finitely many graph charts as in step 2.1. The derivatives $\mathrm{\mid }{\mathrm{\partial }}_{i}H\mathrm{\mid }$ have a positive lower bound on their compact chart supports, and the chart domains, $f$ and ${\mathrm{\Phi }}_t$ are bounded there. Thus the graph integrands have one finite integrable majorant. [F6] gives continuity of both $E\mapsto {\mu }_E(f\circ {\mathrm{\Phi }}_t)$ and $E\mapsto {\mu }_E(f)$ on that subinterval. Their difference integrates to zero against every compactly supported continuous $q$ by step 3.2. If it were nonzero at one energy, continuity would give a neighborhood where it has one strict sign; a nonnegative continuous bump $q$ supported there would make the integral nonzero. Hence the two shell integrals agree for every regular $E\in (a,b)$, including the shell of interest; no flow outside $D$ is used.

Fix a closed subset $F$ of a shell. If $F=\mathrm{\varnothing }$, both the shell measure and its flow pushforward assign it zero. Otherwise, in the compatible band metric from [F9], $F$ is closed and the shell is compactly contained in the band interior. Choose $\delta >0$ so the closed $\delta$-neighborhood of the shell lies in the band, and let $\eta (z)=\max (0,1-\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(z,{\mathrm{\Sigma }}_E)\mathrm{/}\delta )$. It is a continuous cutoff equal to one on the shell and supported in the band. The functions $g_k(z)=\eta (z)\max (0,1-k\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(z,F))$ converge pointwise on the shell to ${\mathbf{1}}_F$ and are bounded by one. Apply the equality proved in step 4.1 to each $g_k$, then [F6] for the finite shell measures gives equality of ${\mu }_E$ and its flow pushforward on every closed subset. Both measures have the same finite total mass by step 3.1; closed sets form a generating $\pi$-system. Apply [F8] and divide by ${\mu }_E(1)>0$ from step 3.1. Thus $P_E$ is invariant for every such regular energy.

The diagonal chart map $F$ has absolute Jacobian ${\prod }_j{\mathrm{\ell }}_j{p}_{0j}=A^n$, so [F4] transforms $c_{N}dqdp$ into ${\widehat{c}}_{N}dQdP$. Applying the graph substitution of step 2.1 in the new coordinates gives its displayed formula with $\tilde{H}$ and ${\widehat{c}}_N$; changing charts back by [F4] gives exactly the same localized shell integrals. Direct substitution in $\lambda ={\sum }_j{p}_{j}d{q}^j$ and ${\omega }_N={\sum }_{j}d{q}^j\wedge d{p}_j$ proves the two pullback formulas. By the convention in [F7], contracting $X_H$ into that form gives the canonical equations ${\dot{q}}^j={\mathrm{\partial }}_{p_j}H$, ${\dot{p}}_j=-{\mathrm{\partial }}_{q^j}H$. Since ${\mathrm{\partial }}_{P_j}\tilde{H}=p_{0j}{\mathrm{\partial }}_{p_j}H$ and ${\mathrm{\partial }}_{Q^j}\tilde{H}={\mathrm{\ell }}_j{\mathrm{\partial }}_{q^j}H$, division of the two velocity components by ${\mathrm{\ell }}_j$ and $p_{0j}$ proves the stated $A^{-1}$ factors in the unchanged parameter $t$.