\subsection{Flux transported by actual arrivals}
\label{sec:arrival-flux-transport}

An arrival time varies with the initial point. Its derivative
adds a vector in the flow direction to the derivative of the
fixed-time map. In the flux determinant this vector repeats
the direction already contracted into the volume form, so
its contribution vanishes. This observation transfers the
multiplier of the preceding subsection to actual crossings.

Let $\tau$ be $C^1$ near $x\in\M$, and suppose
$\tau(x)\in I(x)$. On a smaller neighborhood the arrival
map $A(y)=\phi_{\tau(y)}y$ is defined. The chain rule gives
\begin{equation}
 DA(x)v=D\phi_{\tau(x)}(x)v
                   +d\tau_x(v)F(A(x)).
 \label{eq:moving-arrival-differential}
\end{equation}
The identity $\mathcal C\circ A=\mathcal C$ shows that
$DA(x)$ preserves constraint tangent spaces. For tangent
vectors $v_1,v_2,v_3$,
\begin{equation}
 \omega_{A(x)}(DA(x)v_1,DA(x)v_2,DA(x)v_3)
       =J(x,\tau(x))\omega_x(v_1,v_2,v_3).
 \label{eq:moving-arrival-flux}
\end{equation}
Indeed, after inserting~\eqref{eq:moving-arrival-differential},
every term containing an added $F(A(x))$ vanishes by
alternation. The remaining term is
\eqref{eq:section-flux-transport}.

Suppose an arrival between the charts $\mathcal X_h$ and
$\mathcal X_k$ has coordinate map $R$, so that on an open
neighborhood of $y\in\mathcal D_h$,
\begin{equation}
 \mathcal X_k(R(v))
       =\phi_{\tau(\mathcal X_h(v))}\mathcal X_h(v),
 \qquad R(v)\in\mathcal D_k.
 \label{eq:arrival-chart-identity}
\end{equation}
Assume $h,k>0$ and $R$ is $C^1$. Applying
\eqref{eq:moving-arrival-flux} to the three chart tangent
vectors and using~\eqref{eq:physical-section-signed-flux}
gives
\begin{equation}
 w_k(R(y))|\det DR(y)|
    =J(\mathcal X_h(y),\tau(\mathcal X_h(y)))w_h(y).
 \label{eq:arrival-coordinate-jacobian}
\end{equation}
All factors on the right are positive. Thus $\det DR(y)\ne0$,
and the weighted coordinate Jacobian
\begin{equation}
 j(y)=\frac{w_k(R(y))|\det DR(y)|}{w_h(y)}
       =\exp\left(\int_0^{\tau(\mathcal X_h(y))}
                      \mathcal D(\phi_u\mathcal X_h(y))\dd u\right)
 \label{eq:arrival-weighted-jacobian}
\end{equation}
is strictly greater than one whenever the arrival time is positive.

\begin{proposition}
\label{prop:local-arrival-measure}
Let $h,k>0$, $y\in\mathcal D_h$ and $t\in I(\mathcal X_h(y))$.
Suppose the orbit at time $t$ satisfies $N_1=k$ and $p<0$.
There are an open neighborhood $U\subset\mathcal D_h$
of $y$, a $C^1$ arrival-time function $\tau$ near
$\mathcal X_h(y)$, and a $C^1$ injective map
$R:U\to\mathcal D_k$ such that
$\tau(\mathcal X_h(y))=t$ and
\eqref{eq:arrival-chart-identity} holds on $U$.
In a neighborhood of $(y,t)$ in the space of initial
coordinates and time, the equation $N_1(\phi_u\mathcal X_h(v))=k$
is equivalent to $u=\tau(\mathcal X_h(v))$.

The function $j$ in~\eqref{eq:arrival-weighted-jacobian}
is positive and continuous on $U$. For every Borel
$A\subset U$, the image $R(A)$ is Borel and
\begin{equation}
 \mu_k(R(A))=\int_A j(y)\dd\mu_h(y).
 \label{eq:local-arrival-change-of-variables}
\end{equation}
In particular $\mu_k(R(A))=0$ if and only if $\mu_h(A)=0$.
If $t>0$, $U$ can be chosen so that all arrival times
are positive, and then
\begin{equation}
 0<\mu_h(A)<\infty
       \quad\Longrightarrow\quad
       \mu_h(A)<\mu_k(R(A)).
 \label{eq:local-arrival-strict-expansion}
\end{equation}
\end{proposition}

\begin{proof}
At the given crossing,
\[
 \frac d{du}N_1(\phi_u\mathcal X_h(y))\bigg|_{u=t}
        =(-p^2-s^2+2p)k<0.
\]
The implicit function theorem applied to $N_1(\phi_u x)-k$
therefore provides a $C^1$ function $u=\tau(x)$ and the
stated local uniqueness in initial state and time. Positivity
of the walls and the constraint persist along the physical
flow. At the endpoint, $p<0$ persists by continuity.
The inverse logarithmic coordinates consequently give a
$C^1$ map $R$ satisfying~\eqref{eq:arrival-chart-identity}
on a neighborhood of $y$.

Equation~\eqref{eq:arrival-coordinate-jacobian} makes its
derivative invertible. The inverse function theorem supplies
a smaller open neighborhood on which $R$ is injective.
Restrict further to retain the domain, endpoint and event
properties. If $t>0$, continuity of
$\tau\circ\mathcal X_h$ also retains positive arrival times.

The quotient defining $j$ is continuous on this neighborhood,
because $R$ is $C^1$ and $w_h,w_k$ are positive continuous
densities on their respective chart domains. A Borel subset
of $U$ has Borel image under the continuous injective map
$R$. The change of variables formula gives
\[
 \begin{split}
 \mu_k(R(A))
   &=\int_A w_k(R(y))|\det DR(y)|\dd y\\
   &=\int_A j(y)w_h(y)\dd y
    =\int_A j(y)\dd\mu_h(y).
 \end{split}
\]
The positivity of $j$ proves equivalence of the null sets.
For positive-time arrivals, $j>1$ throughout $U$.
If $0<\mu_h(A)<\infty$, strict monotonicity of the
nonnegative integral gives
$\int_Aj\dd\mu_h>\int_A1\dd\mu_h=\mu_h(A)$.
\end{proof}

This result concerns any transverse arrival on the negative
$p$ branch. To select the first visit to a prescribed Borel
part of that section, we next organize these local maps into
a countable family and use the order of their actual times.