FiniteEnergyPlane.exists_hasWindInfAt_of_hasAsymptoticEigenvector

Data and hypotheses. Let $H:\mathbb{R}^4\to\mathbb{R}$ have a strictly star-shaped level and $J$ be a compatible $\xi$-structure for $H$. Let $z_0\in\mathbb{C}$, $r>0$, and let $u:\mathbb{C}\to\mathbb{R}^4$ be a finite-energy curve on the punctured disk $D^*=\{z: 0<|z-z_0|<r\}$. Let $P=(x,T)$ be a Reeb orbit of $H$, $\varepsilon\in\{1,-1\}$, $\mu\in\mathbb{R}$, and $v:\mathbb{R}\to\mathbb{R}^4$, and assume that $v$ is an asymptotic eigenvector of $u$ at the end $\mathrm{cyl}_{z_0}$ with orbit $P$, sign $\varepsilon$ and eigenvalue $\mu$, which means all of:

• $v$ is an eigenvector of the asymptotic operator of $(J,P)$ with eigenvalue $\mu$ (defined below);
• $\varepsilon\mu<0$;
• there are $c\in\mathbb{R}$, functions $\tau:\mathbb{R}^2\to\mathbb{R}$, $\rho:\mathbb{R}^2\to\mathbb{R}^4$ and $s_0\in\mathbb{R}$ such that $(s,t)\mapsto\tau(s,t)-\varepsilon Tt - c$ and $\rho$ both tend to $0$ in $C^\infty$ as $s\to+\infty$, and for all $s\ge s_0$ and all $t$, setting

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

one has $h(s,t)\in\xi_{x(\tau(s,t))}$ and

$$W_{\mathrm{cyl}_{z_0}}(s,t) = \mathrm{pr}\big(x(\tau(s,t)) + h(s,t)\big).$$

"$g$ tends to $0$ in $C^\infty$ as $s\to+\infty$" (for $g$ on $\mathbb{R}^2$) means: $g$ is $C^\infty$ on $(s_1,\infty)\times\mathbb{R}$ for some $s_1$, and for every order $m\ge 0$ and every $\delta>0$ there is $s_2$ with $\|D^m g(s,t)\|\le\delta$ for all $s\ge s_2$ and all $t$.

Conclusion. There is an integer $k$ such that

1. $\mathrm{wind}_\xi(P, v) = k$, i.e. the loop $t\mapsto\Xi_{x(Tt)}(v(t))$ has $\mathrm{wind} = k$; and
2. $\mathrm{wind}_\infty(u, \mathrm{cyl}_{z_0}, \varepsilon) = k$, i.e. there is $s_0'$ such that for every $s\ge s_0'$ the loop

$$t\mapsto \Xi_{W(s,\varepsilon t)}\Big(\pi_\xi\big(W(s,\varepsilon t)\big)\,\partial_s W(s,\varepsilon t)\Big),\qquad W = W_{\mathrm{cyl}_{z_0}},$$

has $\mathrm{wind} = k$, where $\partial_sW(s,t')$ is the derivative of $W$ at $(s,t')$ in direction $(1,0)$ (for $\varepsilon=-1$ the loop runs with $t$ reversed).

No nondegeneracy, primality or $\mathrm{NdEnd}$ hypothesis on $P$ is assumed; the only link between $u$ and $P$ is the asymptotic-eigenvector hypothesis. The hypotheses on $H$, $J$ and finite energy are as defined below.

Asymptotic eigenvector of $(J,P)$. "$v$ is an eigenvector with eigenvalue $\mu$" means: $v$ is $C^\infty$ and $1$-periodic, $v(t)\in\xi_{x(Tt)}$ for all $t$, $v(t)\neq0$ for some $t$, and for all $t$

$$J\big(x(Tt)\big)\Big(-v'(t) + T\,DR\big(x(Tt)\big)v(t)\Big) = \mu\,v(t).$$

Meaning of the notions used

Throughout, the ambient space is $\mathbb{R}^4$ with coordinates $(y_0,y_1,y_2,y_3)$ (the code's space of functions on a 4-element index set), and $\mathbb{C}$ is identified with $\mathbb{R}^2$ via $z = s + it$.

Linear algebra. $j_0(v) = (-v_1, v_0, -v_3, v_2)$ and

$$\omega_0(u,v) = u_0v_1 - u_1v_0 + u_2v_3 - u_3v_2, \qquad \lambda_x(v) = \tfrac12\,\omega_0(x,v).$$

Star-shaped level. "$H$ has a strictly star-shaped level" means: $H:\mathbb{R}^4\to\mathbb{R}$ is $C^\infty$; for every $u\neq 0$ there is exactly one $r>0$ with $H(ru)=1$; and $dH_x(x) > 0$ at every $x$ with $H(x)=1$. Write $S = \{H = 1\}$.

Reeb field and contact plane. $X_H(x) = j_0(\nabla H(x))$, and $R(x) = \lambda_x(X_H(x))^{-1} X_H(x)$ (with the convention $0^{-1} = 0$). The plane $\xi_x$ is $\{v : \lambda_x(v) = 0 \text{ and } dH_x(v) = 0\}$ (defined for every $x$, not only on $S$). The projection $\pi_\xi$ is $\pi_\xi(x)v = v - \lambda_x(v)\,R(x)$.

Compatible $J$. "$J$ is a compatible $\xi$-structure for $H$" means: $J:\mathbb{R}^4\to \mathrm{L}(\mathbb{R}^4,\mathbb{R}^4)$ is $C^\infty$, and for every $x\in S$ and all $v,w\in\xi_x$: $J(x)v\in\xi_x$, $J(x)^2 v = -v$, $\omega_0(v,J(x)v) > 0$ if $v\neq 0$, and $\omega_0(J(x)v,J(x)w) = \omega_0(v,w)$. Nothing is required of $J(x)$ off $\xi_x$ or off $S$.

Radial projection. For $y\in\mathbb{R}^4$ let $\sigma(y)$ be the unique $r>0$ with $H(ry)=1$ if such $r$ exists, and $\sigma(y)=1$ otherwise (in particular $\sigma(0)=1$). Then $\mathrm{pr}(y) = \sigma(y)\,y$ and $a(y) = -\tfrac12\log\sigma(y)$.

Almost complex structure $\tilde J$. For $y\in\mathbb{R}^4$ put $x = \mathrm{pr}(y)$, $r=\sigma(y)$, $\alpha(v) = \frac{r}{2\,dH_x(x)}\,dH_x(v)$, $\beta(v) = r\,\lambda_x(v)$, $w(v) = v - 2\alpha(v)\,y - \beta(v)\,r^{-1}R(x)$, and

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

Finite-energy curve. For $U\subseteq\mathbb{C}$, "$u$ is a finite-energy curve on $U$" means: $u:\mathbb{C}\to\mathbb{R}^4$ is $C^\infty$ on $U$; $u(z)\neq 0$ for $z\in U$; at every $z\in U$, $u$ is (real-)differentiable and $\partial_s u(z) + \tilde J(u(z))\,\partial_t u(z) = 0$; and

$$E(u) = \sup_{\varphi}\int_U \max\Big(0,\ \varphi'(a)\big(\partial_s a\,\lambda_w(\partial_t w) - \partial_t a\,\lambda_w(\partial_s w)\big) + \varphi(a)\,\omega_0(\partial_s w,\partial_t w)\Big)\,dz < \infty,$$

where $a = a\circ u$, $w = \mathrm{pr}\circ u$, and $\varphi$ ranges over $C^\infty$ nondecreasing functions $\mathbb{R}\to[0,1]$. Values of $u$ outside $U$ are unconstrained. Derivatives here and below are Fréchet derivatives, with the convention that the derivative at a point of non-differentiability is $0$.

Reeb orbit. A Reeb orbit $P$ of $H$ is a pair $(x,T)$ with $T>0$ and $x:\mathbb{R}\to\mathbb{R}^4$ such that $H(x(\tau)) = 1$, $x'(\tau) = R(x(\tau))$ and $x(\tau+T) = x(\tau)$ for all $\tau$. $T$ is any positive period (not necessarily the minimal one).

End parametrizations. $\mathrm{cyl}_\infty(s,t) = e^{2\pi s}e^{2\pi i t}$ (large circles, counterclockwise in $t$), and for $z_0\in\mathbb{C}$, $\mathrm{cyl}_{z_0}(s,t) = z_0 + e^{-2\pi s}e^{-2\pi i t}$ (small circles around $z_0$, clockwise in $t$). For such a map $e$ write $W_e(s,t) = \mathrm{pr}(u(e(s,t)))$ and $A_e(s,t) = a(u(e(s,t)))$. Note that only large $s$ enters the conditions below; for small $s$ these maps may leave the region where $u$ is constrained.

"Nondegenerate end". For $\varepsilon\in\mathbb{R}$, $\mathrm{NdEnd}(u, e, P, \varepsilon)$ means: there are $d, c\in\mathbb{R}$ and $b>0$ such that, uniformly in $t\in\mathbb{R}$, as $s\to+\infty$,

$$A_e(s,t) - \varepsilon T s - d \to 0, \qquad e^{bs}\,\big|W_e(s,t) - x(\varepsilon T t + c)\big| \to 0.$$

(This is an exponential convergence condition on $u$ along the end; it contains no hypothesis on the linearized Reeb flow of $P$.)

$\xi$-coordinates. For $x, v\in\mathbb{R}^4$, with $|x|$ the Euclidean norm, define $Q_1x = (-x_2, x_3, x_0, -x_1)$, $Q_2x = (x_3, x_2, -x_1, -x_0)$,

$$e_i(x) = |x|^{-1}\Big(Q_ix - \frac{dH_x(Q_ix)}{dH_x(x)}\,x\Big)\quad(i=1,2),$$

(with $0^{-1}=0$ and division by $0$ giving $0$), and

$$\Xi_x(v) = \omega_0(v, e_2(x)) + i\,\omega_0(e_1(x), v)\ \in\mathbb{C}.$$

Winding of a loop. For $f:\mathbb{R}\to\mathbb{C}$ and $k\in\mathbb{Z}$, $\mathrm{wind}(f) = k$ means: $f$ is continuous, $1$-periodic, never $0$, and there is a continuous $\theta:\mathbb{R}\to\mathbb{R}$ with $f(t) = |f(t)|e^{i\theta(t)}$ for all $t$ and $\theta(1) - \theta(0) = 2\pi k$.