Read-back: FiniteEnergyPlane.windInf_bounds_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 condition "$v \in \xi_x$" means $\lambda_x(v) = 0$ and $dH_x(v) = 0$.

• Strictly star-shaped $H$: $H$ is $C^\infty$; for every $u \neq 0$ there is exactly one $r>0$ with $H(ru)=1$; and $dH_x(x)>0$ whenever $H(x)=1$.
• Compatible $\xi$-structure $J : \mathbb{R}^4 \to L(\mathbb{R}^4,\mathbb{R}^4)$: $J$ is $C^\infty$, and for every $x\in S$ and $v,w\in\xi_x$ we have $J_xv\in\xi_x$, $J_xJ_xv=-v$, $\omega_0(v,J_xv)>0$ if $v\ne0$, and $\omega_0(J_xv,J_xw)=\omega_0(v,w)$. Nothing is required of $J_x$ off $\xi_x$ or for $x\notin S$.
• Reeb orbit $P = (x, T)$: 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. $P$ is called prime if $x(\tau)\ne x(0)$ for all $0<\tau<T$.
• Rescaling: $\sigma(y)$ is a chosen $r>0$ with $H(ry)=1$ if one exists, and $\sigma(y)=1$ otherwise. For $y\ne0$ this $r$ is unique; for $y=0$ with $H(0)=1$ it is an arbitrary positive choice. Then $\operatorname{proj}(y)=\sigma(y)\,y$ and $a(y)=-\tfrac12\log\sigma(y)$.
• Extended structure $\tilde J$: 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

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

• Finite-energy curve on $\mathbb{C}$: a map $u:\mathbb{C}\to\mathbb{R}^4$ such that

  • $u$ is $C^\infty$ on $\mathbb{C}$ and $u(z)\ne0$ for every $z$;
  • at every $z$, $u$ is differentiable and $\partial_su+\tilde J(u)\,\partial_tu=0$, where $\partial_s$ and $\partial_t$ are the derivatives in the directions $1$ and $i$;
  • $E(u)<\infty$.

  Here, with $a_u=a\circ u$ and $w_u=\operatorname{proj}\circ u$,

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

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

• "Nondegenerate end" at $\infty$ asymptotic to $P$ with $\varepsilon=1$: let $e(s,t)=e^{2\pi s}e^{2\pi it}$ and $W(s,t)=\operatorname{proj}(u(e(s,t)))$. There are $d,c,b\in\mathbb{R}$ with $b>0$ such that, uniformly in $t\in\mathbb{R}$ as $s\to+\infty$,

$$a(u(e(s,t)))-Ts-d\to0\qquad\text{and}\qquad e^{bs}\bigl(W(s,t)-x(Tt+c)\bigr)\to0.$$

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

• Loop winding: a function $f:\mathbb{R}\to\mathbb{C}$ has loop winding $k\in\mathbb{Z}$ if
  • $f$ is continuous, $1$-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 $t$;
  • $\theta(1)-\theta(0)=2\pi k$.
• $\xi$-coordinates: $\xi C(x,v)=\omega_0(v,f_2(x))+i\,\omega_0(f_1(x),v)\in\mathbb{C}$, 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 $Q_2x=(x_3,x_2,-x_1,-x_0)$. Division by $0$ gives $0$. The Reeb-removing map is $\pi_\xi(x,v)=v-\lambda_x(v)R(x)$.

• Winding $k$ at $\infty$: there is $s_0$ such that, for every $s\ge s_0$, the loop $t\mapsto\xi C\bigl(W(s,t),\ \pi_\xi(W(s,t),\partial_sW(s,t))\bigr)$ has loop winding $k$. Here $\partial_sW$ is the Fréchet derivative of $W$ on $\mathbb{R}^2$ applied to $(1,0)$, which is $0$ by convention wherever $W$ is not differentiable.

• $\pi$-winding $k$ (with no punctures): let $w_u=\operatorname{proj}\circ u$ and $D(z)=\xi C\bigl(w_u(z),\ \pi_\xi(w_u(z),\partial_sw_u(z))\bigr)$. The condition says the zero set of $D$ on $\mathbb{C}$ is a finite set $Z$, and there are integers $\operatorname{idx}(z)$, for $z\in Z$, with $\sum_{z\in Z}\operatorname{idx}(z)=k$. Each zero $z_0\in Z$ must have local index $\operatorname{idx}(z_0)$, meaning:

  • there is $r>0$ such that $z_0$ is the only zero of $D$ in the closed disc $\bar B(z_0,r)$;
  • for every $\rho\in(0,r)$, the loop $t\mapsto D(z_0+\rho e^{2\pi it})$ has loop winding $\operatorname{idx}(z_0)$.

  When $k=0$, $Z$ need not be empty: only the sum of the indices is $0$.

• Fast plane $u$: $u$ is a finite-energy curve on $\mathbb{C}$; there is some prime Reeb orbit $P'$ such that $u$ has a nondegenerate end at $\infty$ asymptotic to $P'$ with $\varepsilon=1$ (in the sense above); and $u$ has $\pi$-winding $0$ with no punctures.

Additional notions on $\mathbb{R}^4$ (Hamiltonian side). In these definitions $H_0:\mathbb{R}^4\to\mathbb{R}$, and $\omega_0$, $\lambda$, $Q_1$, $Q_2$ are the same formulas as above.

• $X_{H_0}(y)=(-\partial_1H_0,\ \partial_0H_0,\ -\partial_3H_0,\ \partial_2H_0)(y)$.
• Strictly star-shaped $H_0$: the same three conditions as above, for $H_0$.
• Periodic orbit $P$ of $H_0$: a map $P.x:\mathbb{R}\to\mathbb{R}^4$ and a number $P.T>0$ with $(P.x)'(t)=X_{H_0}(P.x(t))$, $H_0(P.x(t))=1$ and $P.x(t+P.T)=P.x(t)$ for all $t$. It need not be prime.
• Linearized flow $Y$ along $P.x$: a map $Y:\mathbb{R}\to L(\mathbb{R}^4,\mathbb{R}^4)$ with $Y(0)=\mathrm{id}$ and $Y'(t)=D X_{H_0}(P.x(t))\circ Y(t)$ for all $t$. Here $DX_{H_0}$ is the Fréchet derivative, which is $0$ where $X_{H_0}$ is not differentiable.
• Frames: $f^0_1, f^0_2$ are the frames $f_1, f_2$ above, built from $H_0$.
• Projection: $\rho(x,v)=v-\dfrac{\lambda_x(v)}{\lambda_x(X_{H_0}(x))}X_{H_0}(x)$, with division by $0$ giving $0$.
• Coordinates: $c(x,v)=\bigl(\omega_0(v,f^0_2(x)),\ \omega_0(f^0_1(x),v)\bigr)\in\mathbb{R}^2$.
• Linearized $\xi$-path: $\Psi(\tau)$ is the real $2\times2$ matrix whose $j$-th column ($j=1,2$) is

$$c\Bigl(P.x(P.T\,\tau),\ \rho\bigl(P.x(P.T\,\tau),\ Y(P.T\,\tau)\,f^0_j(P.x(0))\bigr)\Bigr).$$

• Index of a path $\varphi:\mathbb{R}\to M_2(\mathbb{R})$:
  • An angle lift from $s$ is a function $\theta$ that is continuous on $[0,1]$, has $\theta(0)=s$, and satisfies: for every $t\in[0,1]$ there is $r>0$ with $\varphi(t)(\cos s,\sin s)=r(\cos\theta(t),\sin\theta(t))$.
  • Let $I$ be the set of numbers $(\theta(1)-s)/(2\pi)$ over all $s\in[0,2\pi]$ and all angle lifts $\theta$ from $s$.
  • Let $a=\inf I$ and $b=\sup I$. By convention, the infimum or supremum of an empty set, or of a set unbounded on that side, is $0$.
  • Then $\operatorname{ind}(\varphi)=2(\lceil b\rceil-1)$ if $a\le\lceil b\rceil-1$, and $\operatorname{ind}(\varphi)=2\lceil b\rceil-1$ otherwise.
• Nondegenerate Reeb orbit $Q=(Q.x,Q.T)$: for every $\Phi:\mathbb{R}\to L(\mathbb{R}^4,\mathbb{R}^4)$ with $\Phi(0)=\mathrm{id}$ and $\Phi'(\tau)=DR(Q.x(\tau))\circ\Phi(\tau)$ for all $\tau$, and every $v\in\xi_{Q.x(0)}$ with $\Phi(Q.T)v=v$, we have $v=0$. This is vacuous if no such $\Phi$ exists. $DR$ is the Fréchet derivative of the Reeb field, which is $0$ where $R$ is not differentiable.

Statement. Let $H_0:\mathbb{R}^4\to\mathbb{R}$ be strictly star-shaped. Let $H=H_0$, viewed on $V=\mathbb{R}^4$ through the identity reindexing of coordinates. Below, $\iota$ denotes this identity identification, with $\iota(y)_k=y_k$. Then for:

• every compatible $\xi$-structure $J$ for $H$;
• every periodic orbit $P$ of $H_0$ and every Reeb orbit $Q=(Q.x,Q.T)$ of $H$ such that:
  • $Q.x(\mathbb{R})=\iota(P.x(\mathbb{R}))$ (equal images; no relation between the parametrizations is required);
  • $Q.T=\displaystyle\int_0^{P.T}\lambda_{\iota P.x(\tau)}\Bigl(\tfrac{d}{d\tau}\iota P.x(\tau)\Bigr)\,d\tau$;
  • $Q$ is nondegenerate (as defined above);
• every finite-energy curve $u$ on $\mathbb{C}$ (for $H, J$) that has a nondegenerate end at $\infty$ asymptotic to $Q$ with $\varepsilon=1$,

there is an integer $k$ such that all of the following hold:

1. $u$ has winding $k$ at $\infty$;
2. $u$ has $\pi$-winding $k-1$ with no punctures (the zero set of $D$ on $\mathbb{C}$ is finite, and its local indices sum to $k-1$);
3. $k\ge1$;
4. for every linearized flow $Y$ along $P.x$,

$$2k\ \le\ \operatorname{ind}(\Psi),$$

where $\Psi$ is the linearized $\xi$-path of $P$ and $Y$.