Exponential decay of the normal component of a nondegenerate end with a drifting phase

Let $S=H^{-1}(1)\subset\mathbb{R}^{2n+2}$ be a strictly star-shaped level with contact planes $\xi$, $J$ a smooth $d\lambda_0$-compatible complex structure on $\xi$, $u$ a finite-energy $\tilde J$-holomorphic curve on $D\setminus\{0\}$, $P=(x,T)$ a periodic Reeb orbit and $\varepsilon=\pm1$. In the cylindrical coordinates $e^{-2\pi(s+it)}$ at $0$ write $a(s,t)$ for the symplectization coordinate and $w(s,t)\in S$ for the radial projection of $u$. Suppose that for $s\ge s_0$

with $\tau(s,t+1)=\tau(s,t)+\varepsilon T$ and $h(s,t+1)=h(s,t)$, where $\tau(s,t)-\varepsilon Tt$ is smooth for large $s$ and every derivative of order $\ge1$ of $\tau(s,t)-\varepsilon Tt$ tends to $0$ as $s\to+\infty$, uniformly in $t$ (no limit of $\tau-\varepsilon Tt$ itself is assumed), and $h\to0$ in $C^\infty$, uniformly in $t$. If $P$ is nondegenerate, then $h$ decays exponentially: there are $\delta>0$, $C$ and $s_1$ with $\|h(s,t)\|\le Ce^{-\delta s}$ for all $s\ge s_1$ and all $t$.

This is the $C^0$ exponential decay of the normal part of an end at a nondegenerate orbit, in the situation where the position $\tau$ along the orbit is not yet known to converge (Hofer–Wysocki–Zehnder 1996, Lemma 2.6).

Formalization Note proj H is the radial projection onto $S$, IsInXi H x v means $v\in\xi_x=\ker\lambda_0(x)\cap\ker dH(x)$, and $C^\infty$ convergence is TendstoZeroCInfty (smooth on some $(s_0,\infty)\times\mathbb{R}$, every derivative tends to $0$ uniformly in $t$). Derivatives are iteratedFDeriv ℝ on $\mathbb{R}\times\mathbb{R}$. $a$ and $w$ are endA H u cylEnd and endW H u cylEnd; norms on $V=\mathbb{R}^{2n+2}$ are sup norms.