Tube coordinates near a periodic Reeb orbit on a star-shaped level

Let $S=H^{-1}(1)\subset\mathbb{R}^{2n+2}$ be a strictly star-shaped level with contact planes $\xi_x=\ker\lambda_0(x)\cap\ker dH(x)$, $P=(x,T)$ a periodic Reeb orbit on $S$, $\varepsilon=\pm1$ and $c\in\mathbb{R}$. Let $w:\mathbb{R}^2\to\mathbb{R}^{2n+2}$ take values in $S$ for $s>s_0$, be $1$-periodic in $t$, and satisfy $w(s,t)-x(\varepsilon Tt+c)\to0$ in $C^\infty$, uniformly in $t$, as $s\to+\infty$. Then there are functions $\tau$, $h$ and constants $s_1$, $K$ such that $\tau(s,t)-\varepsilon Tt-c\to0$ and $h\to0$ in $C^\infty$, and for all $s\ge s_1$ and all $t$:

• $\tau(s,t+1)=\tau(s,t)+\varepsilon T$ and $h(s,t+1)=h(s,t)$;
• $h(s,t)\in\xi_{x(\tau(s,t))}$ and $w(s,t)=\mathrm{proj}\big(x(\tau(s,t))+h(s,t)\big)$;
• $\|w(s,t)-x(\tau(s,t))\|\le K\|h(s,t)\|$ and $\|w(s,t)-x(\varepsilon Tt+c)\|\le K\big(\|h(s,t)\|+|\tau(s,t)-\varepsilon Tt-c|\big)$.

These are the tube coordinates $(\tau,h)$ in which the asymptotic behaviour of a finite-energy curve near a Reeb orbit is described (as in Hryniewicz, Theorem 2.11); $\tau$ is the position along the orbit and $h$ the component normal to it.

Formalization Note proj H is the radial projection onto $S$, IsInXi H x v means $v\in\xi_x$, and $C^\infty$ convergence is TendstoZeroCInfty (smooth on some $(s_0,\infty)\times\mathbb{R}$, every derivative tends to $0$ uniformly in $t$). Norms on $V=\mathbb{R}^{2n+2}$ are sup norms.