Read-back: FiniteEnergyPlane.isIsolated_of_isNondegenerate

Setting. Fix $n \in \mathbb N$ (including $n = 0$) and let $V = \mathbb R^{2n+2}$ with coordinates $v_0, \dots, v_{2n+1}$ and its usual topology. Write $(j_0 v)_{2i} = -v_{2i+1}$, $(j_0 v)_{2i+1} = v_{2i}$, $\langle u, v\rangle = \sum_k u_k v_k$, $\omega_0(u, v) = \langle j_0 u, v\rangle$ and $\lambda_x(v) = \tfrac12\, \omega_0(x, v)$. For $H : V \to \mathbb R$, $dH_x$ is the Fréchet derivative at $x$ (the zero map where $H$ is not differentiable), $\nabla H(x) = (dH_x(e_k))_k$, $X_H = j_0 \nabla H$, and the Reeb field is $R(x) = \lambda_x(X_H(x))^{-1} X_H(x)$ with $0^{-1} = 0$. For $x \in V$, $\xi_x = \{v : \lambda_x(v) = 0,\ dH_x(v) = 0\}$.

A Reeb orbit $Q$ of $H$ is a map $x_Q : \mathbb R \to V$ with a number $T_Q > 0$ such that $H(x_Q(\tau)) = 1$, $x_Q$ is differentiable with $x_Q'(\tau) = R(x_Q(\tau))$, and $x_Q(\tau + T_Q) = x_Q(\tau)$, for all $\tau$. $T_Q$ need not be the minimal period.

Hypotheses.

• $H$ is strictly star-shaped: $H$ is $C^\infty$; for every $u \neq 0$ there is exactly one $r > 0$ with $H(r u) = 1$; and $dH_x(x) > 0$ whenever $H(x) = 1$.
• $P$ is a Reeb orbit of $H$ (with $x_P$, $T_P$).
• $P$ is nondegenerate: for every $\Phi : \mathbb R \to \mathrm L(V,V)$ with $\Phi(0) = \mathrm{id}$ and $\Phi'(\tau) = DR(x_P(\tau)) \circ \Phi(\tau)$ for all $\tau$ ($DR$ the Fréchet derivative of $R$, zero where $R$ is not differentiable), and every $v \in \xi_{x_P(0)}$, $\Phi(T_P) v = v$ implies $v = 0$. (If no such $\Phi$ existed, this hypothesis would hold vacuously.)

Conclusion. $P$ is isolated in the following sense: there is an open set $W \subseteq V$ containing the image $x_P(\mathbb R)$ such that every Reeb orbit $Q$ of $H$ with

satisfies $x_Q(\mathbb R) \subseteq x_P(\mathbb R)$.

Only orbits whose recorded period $T_Q$ equals $T_P$ exactly are constrained; orbits with any other recorded period are not mentioned. The conclusion is an inclusion of images, not equality of parametrisations.