FiniteEnergyPlane.czIndexOfPath_eq_alphaMinus_add_alphaPlus

Throughout, $\mathbb R^4$ has coordinates $x = (x_0,x_1,x_2,x_3)$. Two copies of $\mathbb R^4$ appear (indexed by $\{0,1,2,3\}$ in two syntactically different ways); they are identified coordinate-by-coordinate, and the identification is used silently below.

Standard structures on $\mathbb R^4$.

• $\omega_0(u,v) = u_0v_1 - u_1v_0 + u_2v_3 - u_3v_2$, and the Liouville form $\lambda_x(v) = \tfrac12\,\omega_0(x,v)$.
• For $H : \mathbb R^4 \to \mathbb R$: $\partial_i H(x) = dH(x)\,e_i$ (Fréchet derivative applied to the $i$-th basis vector), and the Hamiltonian vector field $X_H(x) = (-\partial_1H, \partial_0H, -\partial_3H, \partial_2H)(x)$.
• The Reeb field $R_H(x) = \lambda_x(X_H(x))^{-1} X_H(x)$ (with $0^{-1} = 0$).
• $\xi_x = \{ v : \lambda_x(v) = 0 \text{ and } dH(x)v = 0\}$.

Data and hypotheses. Let $H_0 : \mathbb R^4 \to \mathbb R$, and set $H = H_0$ (under the identification above). Assume:

1. $H_0$ is a strictly star-shaped level: $H_0$ is $C^\infty$; for every $u \neq 0$ there is exactly one $r > 0$ with $H_0(ru) = 1$; and $dH_0(x)\,x > 0$ at every $x$ with $H_0(x) = 1$.

Then the theorem asserts: for every map $J : \mathbb R^4 \to \mathcal L(\mathbb R^4, \mathbb R^4)$ (continuous linear maps) such that

2. $J$ is a compatible $\xi$-structure for $H$: $J$ is $C^\infty$, and for every $x$ with $H(x) = 1$ and all $v, w \in \xi_x$: $J(x)v \in \xi_x$, $J(x)J(x)v = -v$, $v \neq 0 \Rightarrow \omega_0(v, J(x)v) > 0$, and $\omega_0(J(x)v, J(x)w) = \omega_0(v,w)$;

for every Hamiltonian periodic orbit $P$ of $H_0$ — a map $P.x : \mathbb R \to \mathbb R^4$ and a number $P.T > 0$ with $\dot{P.x}(t) = X_{H_0}(P.x(t))$ and $H_0(P.x(t)) = 1$ for all $t$, and $P.x(t + P.T) = P.x(t)$ —

and for every Reeb orbit $Q$ of $H$ — a map $Q.x : \mathbb R \to \mathbb R^4$ and $Q.T > 0$ with $H(Q.x(\tau)) = 1$, $\dot{Q.x}(\tau) = R_H(Q.x(\tau))$, and $Q.x(\tau + Q.T) = Q.x(\tau)$ for all $\tau$ —

satisfying

3. $Q.x(\mathbb R) = P.x(\mathbb R)$ (equal images as sets);
4. $Q.T = \displaystyle\int_0^{P.T} \lambda_{P.x(\tau)}\big(\tfrac{d}{d\tau}P.x(\tau)\big)\, d\tau$;
5. $Q$ is nondegenerate: for every $\Phi : \mathbb R \to \mathcal L(\mathbb R^4,\mathbb R^4)$ with $\Phi(0) = \mathrm{id}$ and $\dot\Phi(\tau) = dR_H(Q.x(\tau)) \circ \Phi(\tau)$ for all $\tau$ (here $dR_H$ is the derivative of $R_H$ as a map on all of $\mathbb R^4$), every $v \in \xi_{Q.x(0)}$ with $\Phi(Q.T)v = v$ is $v = 0$ (the condition quantifies over all such $\Phi$, so it holds trivially if none exists);

there exist integers $a, b$ such that

• $a$ is "$\alpha^-$" for $(J,Q)$ (defined below),
• $b$ is "$\alpha^+$" for $(J,Q)$,
• $b = a$ or $b = a + 1$,
• for every $Y : \mathbb R \to \mathcal L(\mathbb R^4,\mathbb R^4)$ with $Y(0) = \mathrm{id}$ and $\dot Y(t) = dX_{H_0}(P.x(t)) \circ Y(t)$ for all $t$, one has $\mathrm{cz}(\Psi_Y) = a + b$, with $\Psi_Y$ and $\mathrm{cz}$ defined below.

Asymptotic eigenvectors. For $\mu \in \mathbb R$, a map $v : \mathbb R \to \mathbb R^4$ is an asymptotic eigenvector of $(J,Q)$ with eigenvalue $\mu$ if $v$ is $C^\infty$, $1$-periodic, $v(t) \in \xi_{Q.x(Q.T\,t)}$ for all $t$, $v(t) \neq 0$ for some $t$, and for all $t$

$$J(Q.x(Q.T\,t))\big(-\dot v(t) + Q.T\cdot dR_H(Q.x(Q.T\,t))\,v(t)\big) = \mu\, v(t).$$

$\xi$-coordinates and winding. Let $Q_1(x) = (-x_2, x_3, x_0, -x_1)$, $Q_2(x) = (x_3, x_2, -x_1, -x_0)$, $|x|$ the Euclidean norm, and for $i = 1,2$

$$f_i^H(x) = |x|^{-1}\Big(Q_{3-i}(x) - \tfrac{dH(x)Q_{3-i}(x)}{dH(x)\,x}\, x\Big),$$

i.e. $f_1^H$ uses $Q_2$ and $f_2^H$ uses $Q_1$ (divisions by zero give $0$). Define

$$\mathrm{xc}_H(x, v) = \big(\omega_0(v, f_2^H(x)),\ \omega_0(f_1^H(x), v)\big) \in \mathbb R^2,$$

also viewed as the complex number $\omega_0(v, f_2^H(x)) + i\,\omega_0(f_1^H(x), v)$. A function $f : \mathbb R \to \mathbb C$ has loop winding $k \in \mathbb Z$ if $f$ is continuous, $1$-periodic, nowhere zero, and there is a continuous $\theta : \mathbb R \to \mathbb R$ with $f(t) = |f(t)|e^{i\theta(t)}$ for all $t$ and $\theta(1) - \theta(0) = 2\pi k$. $(J,Q)$ has eigen-winding $k$ at $\mu$ if (i) at least one asymptotic eigenvector with eigenvalue $\mu$ exists, and (ii) for every asymptotic eigenvector $v$ with eigenvalue $\mu$, the function $t \mapsto \mathrm{xc}_H(Q.x(Q.T\,t), v(t))$ has loop winding $k$.

$\alpha^\pm$. $a$ is $\alpha^-$ if some $\mu < 0$ has eigen-winding $a$, and every $k'$ that is the eigen-winding at some $\mu < 0$ satisfies $k' \le a$. $b$ is $\alpha^+$ if some $\mu \ge 0$ has eigen-winding $b$, and every $k'$ that is the eigen-winding at some $\mu \ge 0$ satisfies $b \le k'$.

The path $\Psi_Y$. With $\Pi_x(v) = v - \dfrac{\lambda_x(v)}{\lambda_x(X_{H_0}(x))}\,X_{H_0}(x)$, define for $\tau \in \mathbb R$ the $2\times2$ matrix (indices $i,j \in \{0,1\}$)

$$\Psi_Y(\tau)_{ij} = \mathrm{xc}_{H_0}\Big(P.x(P.T\tau),\ \Pi_{P.x(P.T\tau)}\big(Y(P.T\tau)\, g_j\big)\Big)_i,\qquad g_0 = f_1^{H_0}(P.x(0)),\ g_1 = f_2^{H_0}(P.x(0)).$$

The integer $\mathrm{cz}(\varphi)$ of a path $\varphi : \mathbb R \to \mathbb R^{2\times2}$: with $e(s) = (\cos s, \sin s)$, call $\theta : \mathbb R \to \mathbb R$ an angle lift from $s$ if $\theta$ is continuous on $[0,1]$, $\theta(0) = s$, and for each $t \in [0,1]$ some $r > 0$ has $\varphi(t)e(s) = r\,e(\theta(t))$. Let $W = \{(\theta(1) - s)/2\pi : s \in [0,2\pi],\ \theta \text{ an angle lift from } s\}$, $\alpha = \inf W$, $\beta = \sup W$ (real inf/sup, equal to $0$ by convention if $W$ is empty or unbounded on that side). Then $\mathrm{cz}(\varphi) = 2(\lceil\beta\rceil - 1)$ if $\alpha \le \lceil\beta\rceil - 1$, and $2\lceil\beta\rceil - 1$ otherwise.

Note that the integers $a, b$ are chosen once and the final equation must hold for every solution $Y$ of the linearized Hamiltonian equation along $P.x$; the $\alpha^\pm$ conditions refer only to $J$ and $Q$, while $\Psi_Y$ is built from $H_0$, $P$ and $Y$ alone.