A nondegenerate end at a puncture $z_0$, read at the puncture $0$ of the unit disc

Let $S=H^{-1}(1)\subset\mathbb{R}^{2n+2}$, let $P=(x,T)$ be a periodic Reeb orbit on $S$ and $\varepsilon\in\{\pm1\}$. Let $u:\mathbb{C}\to\mathbb{R}^{2n+2}$, $z_0\in\mathbb{C}$ and $r>0$. Suppose that in the cylindrical coordinates $(s,t)\mapsto z_0+e^{-2\pi(s+it)}$ at $z_0$, $u$ has a nondegenerate end over $P$ with sign $\varepsilon$ (Hryniewicz, Definition 2.9 (1), (2), (4)): for some $d,c\in\mathbb{R}$ and $b>0$, $a(s,t)-\varepsilon Ts-d\to0$ and $e^{bs}\big(w(s,t)-x(\varepsilon Tt+c)\big)\to0$ uniformly in $t$ as $s\to+\infty$, where $a$ and $w$ are the $\mathbb{R}$- and $S$-components of $u$. Then $\tilde u(z)=u(z_0+rz)$ has a nondegenerate end over the same $P$ with the same sign $\varepsilon$ in the cylindrical coordinates $(s,t)\mapsto e^{-2\pi(s+it)}$ at $0$.

Together with the corresponding statement for the curve itself, this moves results about nondegenerate punctures at $0$ of the unit disc to any puncture $z_0$.

Formalization Note $\mathbb{R}^{2n+2}$ is V n; the nondegenerate end is IsNondegenerateEnd H u e P ε with e = cylEndAt z₀ (hypothesis) and e = cylEnd (conclusion). The constants $d,c,b$ of the conclusion need not equal those of the hypothesis.