Read-back: FiniteEnergyPlane.plane_asymptotic_eigenvector

Ambient space and standard structures. Here $n \ge 0$ is an arbitrary natural number and $V = \mathbb{R}^{2(n+1)}$, with coordinates $v_0, \dots, v_{2n+1}$. $V$ carries the sup norm $\|v\| = \max_k |v_k|$ (this is the norm in every estimate below on $V$-valued quantities). Define:

• $j_0 v$ by $(j_0 v)_k = -v_{k+1}$ for even $k$ and $(j_0 v)_k = v_{k-1}$ for odd $k$;
• $\langle u, v \rangle = \sum_k u_k v_k$, $\ \omega_0(u, v) = \langle j_0 u, v \rangle$, and $\lambda_x(v) = \tfrac12\, \omega_0(x, v)$;
• for $H : V \to \mathbb{R}$: $\nabla H(x) = (dH(x) e_k)_k$, $\ X_H(x) = j_0 \nabla H(x)$, and $R_H(x) = \lambda_x(X_H(x))^{-1}\, X_H(x)$, with the convention $0^{-1} = 0$ (so $R_H(x) = 0$ wherever $\lambda_x(X_H(x)) = 0$);
• $\xi_x = \{ v \in V : \lambda_x(v) = 0 \text{ and } dH(x) v = 0 \}$.

Throughout, $dH(x)$, $DR_H(x)$, $\partial_s$, $\partial_t$, $D^m$ denote Fréchet derivatives, with the convention that the derivative of a function at a point where it is not differentiable is $0$.

Hypothesis on $H$ ("strictly star-shaped level"). $H$ is $C^\infty$; for every $y \ne 0$ there is exactly one $r > 0$ with $H(r y) = 1$; and $dH(x)\,x > 0$ whenever $H(x) = 1$.

Hypothesis on $J$ ("compatible $\xi$-structure"). $J$ is a map from $V$ to continuous linear maps $V \to V$; it is $C^\infty$, and for every $x$ with $H(x) = 1$ and all $v, w \in \xi_x$: $J(x) v \in \xi_x$; $J(x) J(x) v = -v$; $\omega_0(v, J(x) v) > 0$ if $v \ne 0$; and $\omega_0(J(x) v, J(x) w) = \omega_0(v, w)$. Nothing is required of $J(x)$ outside $\xi_x$ or at points with $H(x) \ne 1$.

Radial coordinates. For $y \in V$, let $r(y)$ be a fixed (choice-selected) $r > 0$ with $H(r y) = 1$ if such $r$ exists, and $r(y) = 1$ otherwise. Put $\mathrm{pr}(y) = r(y)\, y$ and $a(y) = -\tfrac12 \log r(y)$.

Finite-energy curves. For $f$ defined on $\mathbb{C} = \mathbb{R}^2$ write $\partial_s f(z) = Df(z)\cdot 1$ and $\partial_t f(z) = Df(z)\cdot i$. For $y \in V$ let $x = \mathrm{pr}(y)$, $r = r(y)$, $\alpha(v) = \dfrac{r}{2\, dH(x) x}\, dH(x) v$, $\beta(v) = r\, \lambda_x(v)$, $w(v) = v - 2\alpha(v)\, y - \beta(v)\, r^{-1} R_H(x)$, and

$$\tilde J(y) v = \alpha(v)\, r^{-1} R_H(x) - 2 \beta(v)\, y + J(x)\, w(v).$$

For $u : \mathbb{C} \to V$ put $a_u = a \circ u$ and $w_u = \mathrm{pr} \circ u$. A function $\varphi : \mathbb{R} \to \mathbb{R}$ is admissible if it is $C^\infty$, nondecreasing, and takes values in $[0, 1]$. For $U \subseteq \mathbb{C}$ the energy is the value in $[0, \infty]$

$$E_U(u) = \sup_{\varphi \text{ admissible}} \int_U \max\{0, e_\varphi(z)\}\, dz, \qquad e_\varphi = \varphi'(a_u)\big(\partial_s a_u\, \lambda_{w_u}(\partial_t w_u) - \partial_t a_u\, \lambda_{w_u}(\partial_s w_u)\big) + \varphi(a_u)\, \omega_0(\partial_s w_u, \partial_t w_u),$$

the integral being the Lebesgue integral of a nonnegative extended-real function. "$u$ is a finite-energy curve on $U$" means: $u$ is $C^\infty$ on $U$; $u(z) \ne 0$ for all $z \in U$; at every $z \in U$, $u$ is differentiable and $\partial_s u(z) + \tilde J(u(z))\, \partial_t u(z) = 0$; and $E_U(u) < \infty$.

Reeb orbits. A Reeb orbit $P$ of $H$ consists of a map $x_P : \mathbb{R} \to V$ and a number $T_P > 0$ such that, for all $\tau$, $H(x_P(\tau)) = 1$, $x_P$ has derivative $R_H(x_P(\tau))$ at $\tau$, and $x_P(\tau + T_P) = x_P(\tau)$. ($T_P$ is a period of $x_P$, not required to be the minimal one.)

Nondegenerate Reeb orbit. "$P$ is nondegenerate" means: for every map $\Phi : \mathbb{R} \to \{\text{continuous linear maps } V \to V\}$ with $\Phi(0) = \mathrm{id}$ and $\Phi'(\tau) = DR_H(x_P(\tau)) \circ \Phi(\tau)$ for all $\tau \in \mathbb{R}$, and every $v \in \xi_{x_P(0)}$ with $\Phi(T_P) v = v$, one has $v = 0$. (The condition quantifies over all such $\Phi$; if none exists it holds vacuously.)

Loops and $C^\infty$ convergence of loops. $\gamma^{\mathrm{cw}}_\rho(t) = \rho\, e^{-2\pi i t}$ and $\gamma^{\mathrm{ccw}}_R(t) = R\, e^{2\pi i t}$. For $\gamma : \mathbb{R} \to \mathbb{C}$ the loop of $u$ along $\gamma$ is $t \mapsto \mathrm{pr}(u(\gamma(t)))$. A sequence $f_k : \mathbb{R} \to V$ converges in $C^\infty$ to $g$ if for every $m \in \mathbb{N}$ the $m$-th derivatives $f_k^{(m)}$ converge to $g^{(m)}$ uniformly on all of $\mathbb{R}$ as $k \to \infty$ (with the zero-derivative convention).

Decay notions on half-cylinders. For $f : \mathbb{R}^2 \to F$ (with $F = \mathbb{R}$ or $F = V$), $D^m f(s, t)$ is the $m$-th Fréchet derivative, measured in the operator norm of $m$-linear maps, where $\mathbb{R}^2$ carries the max norm $\|(s, t)\| = \max(|s|, |t|)$; $D^0 f = f$.

• $f \to 0$ uniformly: for every $\epsilon > 0$ there is $s_0$ with $\|f(s, t)\| \le \epsilon$ for all $s \ge s_0$ and all $t \in \mathbb{R}$.
• $f \to 0$ in $C^\infty$: there is $s_0$ such that $f$ is $C^\infty$ on $(s_0, \infty) \times \mathbb{R}$, and for every $m \in \mathbb{N}$ and every $\epsilon > 0$ there is $s_1$ with $\|D^m f(s, t)\| \le \epsilon$ for all $s \ge s_1$, $t \in \mathbb{R}$.
• $f$ decays at rate $b$ in $C^\infty$: there is $s_0$ such that $f$ is $C^\infty$ on $(s_0, \infty) \times \mathbb{R}$, and for every $m \in \mathbb{N}$ there are reals $C_m, s_m$ with $\|D^m f(s, t)\| \le C_m\, e^{-b s}$ for all $s \ge s_m$, $t \in \mathbb{R}$.

End coordinates. $e_0(s, t) = e^{-2\pi s} e^{-2\pi i t} \in \mathbb{C}$ (it tends to the puncture $0$ as $s \to +\infty$) and $e_\infty(s, t) = e^{2\pi s} e^{2\pi i t}$. For an end map $e \in \{e_0, e_\infty\}$ put $A(s, t) = a(u(e(s, t)))$ and $W(s, t) = \mathrm{pr}(u(e(s, t)))$.

• Asymptotically cylindrical end (for $e$, $P$, a real $\varepsilon$): there exist $d, c \in \mathbb{R}$ such that $A(s, t) - \varepsilon T_P s - d \to 0$ in $C^\infty$ and $W(s, t) - x_P(\varepsilon T_P t + c) \to 0$ in $C^\infty$.
• Exponential asymptotics at rate $b$ (for $e$, $P$, $\varepsilon$, $b$): there exist $d, c \in \mathbb{R}$ such that $A(s, t) - \varepsilon T_P s - d$ and $W(s, t) - x_P(\varepsilon T_P t + c)$ both decay at rate $b$ in $C^\infty$.
• Nondegenerate end (for $e$, $P$, $\varepsilon$): there exist $d, c, b \in \mathbb{R}$ with $b > 0$ such that $A(s, t) - \varepsilon T_P s - d \to 0$ uniformly and $e^{b s}\big(W(s, t) - x_P(\varepsilon T_P t + c)\big) \to 0$ uniformly.

Asymptotic eigenvector. For $\eta : \mathbb{R} \to V$ define

$$(\mathbf{A}\eta)(t) = J(x_P(T_P t))\Big(-\eta'(t) + T_P\, DR_H(x_P(T_P t))\, \eta(t)\Big).$$

For $\mu \in \mathbb{R}$, $v : \mathbb{R} \to V$, an end map $e$ and a real $\varepsilon$, "$(\mu, v)$ is an asymptotic eigenvector of the end" means all of:

1. $v$ is $C^\infty$, $1$-periodic, $v(t) \in \xi_{x_P(T_P t)}$ for all $t$, $v(t) \ne 0$ for at least one $t$, and $(\mathbf{A} v)(t) = \mu\, v(t)$ for all $t$;
2. $\varepsilon \mu < 0$;
3. there exist $c \in \mathbb{R}$, $\tau : \mathbb{R}^2 \to \mathbb{R}$, $r : \mathbb{R}^2 \to V$ and $s_0 \in \mathbb{R}$ such that $\tau(s, t) - \varepsilon T_P t - c \to 0$ in $C^\infty$, $r \to 0$ in $C^\infty$, and for all $s \ge s_0$ and all $t \in \mathbb{R}$, with

$$h(s, t) = e^{\varepsilon \mu s}\Big(v\big(\tau(s, t)/T_P\big) + r(s, t)\Big),$$

one has $h(s, t) \in \xi_{x_P(\tau(s, t))}$ and $W(s, t) = \mathrm{pr}\big(x_P(\tau(s, t)) + h(s, t)\big)$.

Statement. Let $n \in \mathbb{N}$, $H : V \to \mathbb{R}$ a strictly star-shaped level, $J$ a compatible $\xi$-structure for $H$, and $u : \mathbb{C} \to V$ a finite-energy curve on all of $\mathbb{C}$ (so $u$ is $C^\infty$, nowhere $0$, satisfies $\partial_s u + \tilde J(u) \partial_t u = 0$ everywhere, and $E_{\mathbb{C}}(u) < \infty$). Assume $u$ is non-constant: $u(z_1) \ne u(z_2)$ for some $z_1, z_2$. Let $P$ be a nondegenerate Reeb orbit of $H$. Let $(R_k)_{k \in \mathbb{N}}$ be real numbers with $R_k \to +\infty$ (no positivity is required for individual $k$), let $c \in \mathbb{R}$, and assume the loops

$$t \mapsto \mathrm{pr}\big(u(R_k e^{2\pi i t})\big)$$

converge in $C^\infty$ (all derivatives, uniformly on $\mathbb{R}$) to $t \mapsto x_P(T_P t + c)$.

Then, for the end map $e_\infty(s,t) = e^{2\pi s} e^{2\pi i t}$ (so $A = a \circ u \circ e_\infty$, $W = \mathrm{pr} \circ u \circ e_\infty$), the orbit $P$ and the sign $\varepsilon = 1$:

1. there exists $b > 0$ such that the end has exponential asymptotics at rate $b$, i.e. for some $d, c'$ both $A(s,t) - T_P s - d$ and $W(s,t) - x_P(T_P t + c')$ decay at rate $b$ in $C^\infty$;
2. there exist $\mu \in \mathbb{R}$ and $v : \mathbb{R} \to V$ such that $(\mu, v)$ is an asymptotic eigenvector of this end; in particular (since $\varepsilon = 1$) $\mu < 0$, and $h(s,t) = e^{\mu s}\big(v(\tau(s,t)/T_P) + r(s,t)\big)$ in item 3 of that definition.

The phases $c'$ in item 1 and $c$ in the eigenvector definition are existentially quantified and are not tied to the $c$ of the hypothesis or to each other.