The asymptotic-operator expression of a contact-plane section stays in the contact plane

Let $S=H^{-1}(1)\subset\mathbb{R}^{2n+2}$ be a strictly star-shaped level with contact planes $\xi$, $P=(x,T)$ a periodic Reeb orbit, and $\eta:\mathbb{R}\to\mathbb{R}^{2n+2}$ differentiable with $\eta(t)\in\xi_{x(Tt)}$ for all $t$. Then for all $t$,

This is the expression to which $J$ is applied in the asymptotic operator $A_P\eta=J(-\eta'+T\,DR\,\eta)$, so the statement says that $A_P$ is well defined on sections of $\xi$ along $P$.

Formalization Note V n is $\mathbb{R}^{2n+2}$, IsStrictlyStarShapedLevel H says $H$ is smooth and $S=H^{-1}(1)$ is strictly star-shaped with $dH(x)x>0$ on $S$, reeb H is the Reeb field $R=X_H/\lambda_0(X_H)$, liouville x v is $\lambda_0(x)v=\tfrac12\omega_0(x,v)$, IsInXi H x v means $v\in\xi_x=\ker\lambda_0(x)\cap\ker dH(x)$, and a ReebOrbit H is $P=(x,T)$ with $x'=R(x)$, $x(\mathbb{R})\subset S$, $x(\tau+T)=x(\tau)$, $T>0$.