Let $X$ be a smooth manifold, let $U \subset X$ be open, let $m \in \mathbb{N}$, and let $P: \mathcal{D}'(U) \to \mathcal{D}'(U)$ be a scalar differential operator of order $m$ with smooth coefficients. Let $p_m \in C^\infty(T^*U \setminus 0; \mathbb{R})$ be the real principal symbol of $P$. Fix a smooth time direction on $U$, meaning that in each adapted coordinate chart $(t,x_1,\dots,x_{n-1})$ the coordinate $t$ is the time coordinate and a cotangent vector is written as $(t,x,\tau,\xi)$, with $\tau \in \mathbb{R}$ dual to $t$ and $\xi \in \mathbb{R}^{n-1}$ dual to $x=(x_1,\dots,x_{n-1})$. Suppose that $P$ is strictly hyperbolic with respect to this time direction: in every adapted coordinate chart, for every $(t,x)$ in the chart domain and every spatial covector $\xi \in \mathbb{R}^{n-1} \setminus \{0\}$, the polynomial $\tau \mapsto p_m(t,x,\tau,\xi)$ has exactly $m$ distinct real roots.

Let

\begin{align*} \operatorname{Char}(P) := \{(x,\xi) \in T^*U \setminus 0 : p_m(x,\xi)=0\}. \end{align*}

Let $\omega$ be the canonical symplectic form on $T^*U \setminus 0$, and let $H_{p_m}: T^*U \setminus 0 \to T(T^*U \setminus 0)$ be the Hamiltonian vector field defined by $\omega(H_{p_m},V)=dp_m(V)$ for every tangent vector $V$. A null bicharacteristic arc of $p_m$ means a smooth integral curve $\gamma: I \to \operatorname{Char}(P)$, where $I \subset \mathbb{R}$ is an interval and $\gamma'(s)=H_{p_m}(\gamma(s))$ for all $s \in I$.

Let $\Gamma \subset T^*U \setminus 0$ be an open conic subset on which $\operatorname{Char}(P) \cap \Gamma$ is a disjoint union of smooth connected characteristic sheets $\Sigma$, and assume that $dp_m \neq 0$ at every point of each such sheet. For every $u \in \mathcal{D}'(U)$, for every characteristic sheet $\Sigma \subset \Gamma$, and for every connected component

\begin{align*} C \subset \Sigma \setminus WF(Pu), \end{align*}

the set

\begin{align*} (WF(u) \setminus WF(Pu)) \cap C \end{align*}

is invariant under the Hamiltonian flow of $p_m$ as long as the flow remains in $C$: if $\gamma: I \to C$ is a null bicharacteristic arc and $\gamma(s_0) \in (WF(u) \setminus WF(Pu)) \cap C$ for some $s_0 \in I$, then $\gamma(s) \in (WF(u) \setminus WF(Pu)) \cap C$ for every $s \in I$. Equivalently, $(WF(u) \setminus WF(Pu)) \cap C$ is a union of connected null bicharacteristic arcs of $H_{p_m}$ contained in $C$.