Read-back: FiniteEnergyPlane.one_le_windInf_of_plane

Setting. Here $V = \mathbb{R}^4$ with coordinates $(v_0,v_1,v_2,v_3)$, and

$$j_0(v) = (-v_1, v_0, -v_3, v_2),\qquad \omega_0(u,v) = \langle j_0 u, v\rangle = u_0v_1 - u_1v_0 + u_2v_3 - u_3v_2,\qquad \lambda_x(v) = \tfrac12\,\omega_0(x,v).$$

The gradient of $H$ is taken coordinatewise, $X_H(x) = j_0(\nabla H(x))$, and the Reeb field is $R(x) = \lambda_x(X_H(x))^{-1} X_H(x)$. If $\lambda_x(X_H(x)) = 0$, the inverse is $0$ by convention, so $R(x) = 0$. The energy surface is $S = \{H = 1\}$. The contact plane condition "$v \in \xi_x$" means $\lambda_x(v) = 0$ and $dH_x(v) = 0$.

Data and hypotheses. Let:

• $H : \mathbb{R}^4 \to \mathbb{R}$ be strictly star-shaped: $H$ is $C^\infty$; for every $u \neq 0$ there is exactly one $r > 0$ with $H(ru) = 1$; and $dH_x(x) > 0$ whenever $H(x) = 1$.
• $J : \mathbb{R}^4 \to L(\mathbb{R}^4,\mathbb{R}^4)$ be a compatible $\xi$-structure: $J$ is $C^\infty$, and for every $x \in S$ and every $v, w \in \xi_x$ we have $J_x v \in \xi_x$, $J_x J_x v = -v$, $\omega_0(v, J_x v) > 0$ if $v \neq 0$, and $\omega_0(J_xv, J_xw) = \omega_0(v,w)$. Nothing is required of $J_x$ off $\xi_x$ or for $x \notin S$.
• $u : \mathbb{C} \to \mathbb{R}^4$ be a finite-energy curve on all of $\mathbb{C}$. This means:
  • $u$ is $C^\infty$ on $\mathbb{C}$ and $u(z) \neq 0$ for every $z$;
  • $u$ is $\tilde J$-holomorphic at every $z$: $u$ is differentiable there and $\partial_s u + \tilde J(u)\,\partial_t u = 0$, where $\partial_s$ and $\partial_t$ are the derivatives in the directions $1$ and $i$;
  • $E(u) < \infty$, with $E$ and $\tilde J$ as defined below.
• $P$ be a Reeb orbit of $H$: a map $x : \mathbb{R} \to \mathbb{R}^4$ and a number $T > 0$ with $x(\tau) \in S$, $x'(\tau) = R(x(\tau))$ and $x(\tau + T) = x(\tau)$ for all $\tau$. The period $T$ need not be minimal, and $P$ is not assumed prime or nondegenerate.
• $u$ have a "nondegenerate end" at $\infty$ asymptotic to $P$ with sign $\varepsilon = 1$; see "Ends" below.

Auxiliary notions.

• $\sigma(y)$ is a chosen $r > 0$ with $H(ry) = 1$ if such an $r$ exists, and $\sigma(y) = 1$ otherwise. For $y \neq 0$ this $r$ is unique. For $y = 0$ with $H(0) = 1$, it is an arbitrary positive choice.
• $\operatorname{proj}(y) = \sigma(y)\,y$ and $a(y) = -\tfrac12 \log \sigma(y)$.
• $\tilde J(y)$ is defined as follows. Put $x = \operatorname{proj}(y)$, $r = \sigma(y)$, $\alpha(v) = r\,dH_x(v)/(2\,dH_x(x))$, $\beta(v) = r\lambda_x(v)$ and $w(v) = v - \alpha(v)\,2y - \beta(v)\,r^{-1}R(x)$. Then

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

• Write $a_u = a\circ u$ and $w_u = \operatorname{proj}\circ u$. Then

$$E(u) = \sup_{\varphi} \int_{\mathbb{C}} \max\bigl(0,\ \varphi'(a_u)\bigl(\partial_s a_u\,\lambda_{w_u}(\partial_t w_u) - \partial_t a_u\,\lambda_{w_u}(\partial_s w_u)\bigr) + \varphi(a_u)\,\omega_0(\partial_s w_u, \partial_t w_u)\bigr)\,dz.$$

The supremum runs over $C^\infty$ monotone $\varphi : \mathbb{R}\to[0,1]$. The integral is a lower Lebesgue integral with values in $[0,\infty]$, so the integrand is the positive part of the density.

Ends. Let $e(s,t) = e^{2\pi s}e^{2\pi i t}$ and $W(s,t) = \operatorname{proj}(u(e(s,t)))$. The condition "nondegenerate end at $\infty$ asymptotic to $P$ with $\varepsilon = 1$" means that there are $d, c, b \in \mathbb{R}$ with $b > 0$ such that, uniformly in $t \in \mathbb{R}$ as $s \to +\infty$,

$$a(u(e(s,t))) - T s - d \to 0 \qquad\text{and}\qquad e^{bs}\bigl(W(s,t) - x(Tt + c)\bigr) \to 0.$$

This condition is only exponential uniform convergence of the end. It says nothing about the linearized return map of $P$.

Winding. A function $f : \mathbb{R}\to\mathbb{C}$ has loop winding $k \in \mathbb{Z}$ if all of the following hold:

• $f$ is continuous and $1$-periodic, and never zero;
• there is a continuous $\theta : \mathbb{R}\to\mathbb{R}$ with $f(t) = |f(t)|\,e^{i\theta(t)}$ for all $t$;
• $\theta(1) - \theta(0) = 2\pi k$.

The $\xi$-coordinate map $\xi C(x, v) \in \mathbb{C}$ is

$$\xi C(x,v) = \omega_0(v, f_2(x)) + i\,\omega_0(f_1(x), v),$$

with frames

$$f_1(x) = |x|^{-1}\Bigl(Q_2x - \tfrac{dH_x(Q_2x)}{dH_x(x)}\,x\Bigr),\qquad f_2(x) = |x|^{-1}\Bigl(Q_1x - \tfrac{dH_x(Q_1x)}{dH_x(x)}\,x\Bigr),$$

where $Q_1x = (-x_2, x_3, x_0, -x_1)$ and $Q_2x = (x_3, x_2, -x_1, -x_0)$. As usual, $|x|$ is the Euclidean norm, and division by $0$ gives $0$.

The Reeb-removing map is $\pi_\xi(x,v) = v - \lambda_x(v)\,R(x)$.

"$u$ has winding $k$ at $\infty$" means: there is $s_0$ such that for every $s \ge s_0$, the loop

$$t \mapsto \xi C\bigl(W(s,t),\ \pi_\xi(W(s,t),\ \partial_s W(s,t))\bigr)$$

has loop winding $k$. Here $\partial_s W$ is the Fréchet derivative of $W$ on $\mathbb{R}^2$ applied to $(1,0)$, which is $0$ by convention wherever $W$ is not differentiable.

Conclusion. Under all these hypotheses, there is an integer $k$ with

$$k \ge 1$$

such that $u$ has winding $k$ at $\infty$ in the sense just described.