Hofer–Wysocki–Zehnder: strictly convex energy surfaces are dynamically convex

Read-back. Let $H : \mathbb{R}^4 \to \mathbb{R}$ be any function. Coordinates on $\mathbb{R}^4$ are written $x = (x_0, x_1, x_2, x_3)$. $DH(x)$ is the Fréchet derivative of $H$ at $x$ (taken to be $0$ where $H$ is not differentiable) and $D^2H(x)$ the derivative of $x \mapsto DH(x)$. The level set is $\Sigma = \{x : H(x) = 1\}$.

Hypothesis $hS$ (strictly star-shaped level). All three of:

1. $H$ is $C^\infty$ on all of $\mathbb{R}^4$;
2. for every $u \in \mathbb{R}^4$ with $u \neq 0$ there is exactly one real $r > 0$ with $H(r u) = 1$;
3. for every $x$ with $H(x) = 1$: $DH(x)\,x > 0$ (the derivative of $H$ at $x$ applied to the position vector $x$ itself is positive).

Hypothesis $hC$ (strictly convex level). For every $x$ with $H(x) = 1$ and every $v \neq 0$ with $DH(x)\,v = 0$:

$$D^2H(x)(v, v) > 0 .$$

Conclusion (dynamical convexity). For every periodic orbit $P$ of $H$ and every $Y$ as below, the integer $\mu(M_{P,Y}) \ge 3$, with all objects unfolded as follows.

Hamiltonian vector field. With $\partial_i H(x) = DH(x)\,e_i$ (standard basis vector $e_i$),

$$X_H(x) = \bigl(-\partial_1 H(x),\ \partial_0 H(x),\ -\partial_3 H(x),\ \partial_2 H(x)\bigr).$$

Periodic orbit. $P$ consists of a curve $x : \mathbb{R} \to \mathbb{R}^4$ and a real $T > 0$ with: $x$ differentiable at every $t$ with $x'(t) = X_H(x(t))$; $H(x(t)) = 1$ for all $t$; and $x(t+T) = x(t)$ for all $t$. $T$ need not be the minimal period: iterated (multiply covered) orbits are included. (Hypothesis $hS$(3) gives $DH \neq 0$ on $\Sigma$, hence $X_H \neq 0$ on $\Sigma$.)

Linearized flow. $Y : \mathbb{R} \to \mathrm{L}(\mathbb{R}^4, \mathbb{R}^4)$ is any map with $Y(0) = \mathrm{id}$ and, for every real $t$, $Y$ differentiable at $t$ with

$$Y'(t) = DX_H(x(t)) \circ Y(t) .$$

The statement quantifies over all such $Y$.

Auxiliary forms and frame. For $u, v, x \in \mathbb{R}^4$:

$$\omega_0(u,v) = u_0v_1 - u_1v_0 + u_2v_3 - u_3v_2, \qquad \lambda_x(v) = \tfrac12\bigl(x_0v_1 - x_1v_0 + x_2v_3 - x_3v_2\bigr),$$

$|x| = \sqrt{x_0^2+x_1^2+x_2^2+x_3^2}$ (Euclidean norm), and

$$Q_1 x = (-x_2,\ x_3,\ x_0,\ -x_1), \qquad Q_2 x = (x_3,\ x_2,\ -x_1,\ -x_0),$$ $$\xi_1(x) = |x|^{-1}\Bigl(Q_2x - \frac{DH(x)\,Q_2x}{DH(x)\,x}\,x\Bigr), \qquad \xi_2(x) = |x|^{-1}\Bigl(Q_1x - \frac{DH(x)\,Q_1x}{DH(x)\,x}\,x\Bigr).$$

Projection along $X_H$ and coordinates:

$$\pi_x(v) = v - \frac{\lambda_x(v)}{\lambda_x(X_H(x))}\,X_H(x), \qquad c_x(v) = \bigl(\omega_0(v, \xi_2(x)),\ \omega_0(\xi_1(x), v)\bigr) \in \mathbb{R}^2 .$$

(Division by zero and $0^{-1}$ evaluate to $0$; on $\Sigma$, $hS$(3) makes $DH(x)\,x > 0$ and forces $x \neq 0$.)

Matrix path. For $\tau \in \mathbb{R}$, $M_{P,Y}(\tau)$ is the $2\times 2$ matrix whose $(i,j)$ entry ($i,j \in \{1,2\}$) is

$$M_{P,Y}(\tau)_{ij} = \Bigl[c_{x(T\tau)}\bigl(\pi_{x(T\tau)}\bigl(Y(T\tau)\,\xi_j(x(0))\bigr)\bigr)\Bigr]_i .$$

So column $j$ is: take the frame vector $\xi_j$ at the initial point $x(0)$, push it by $Y(T\tau)$, project along $X_H$ at $x(T\tau)$, and read off the two coordinates $c$ at $x(T\tau)$.

Index. For a matrix path $\varphi$, let $W(\varphi)$ be the set of $d \in \mathbb{R}$ such that there are $s \in [0,2\pi]$ and $\theta : \mathbb{R} \to \mathbb{R}$, continuous on $[0,1]$, with $\theta(0) = s$, such that for each $t \in [0,1]$ some $r > 0$ satisfies $\varphi(t)(\cos s, \sin s) = r(\cos\theta(t), \sin\theta(t))$, and $d = (\theta(1) - s)/(2\pi)$. Put $a = \inf W(\varphi)$, $b = \sup W(\varphi)$ (with $\inf$/$\sup$ of an empty or unbounded set equal to $0$) and $\lceil b\rceil$ the integer ceiling. Then

$$\mu(\varphi) = \begin{cases} 2(\lceil b\rceil - 1) & \text{if } a \le \lceil b\rceil - 1,\\ 2\lceil b\rceil - 1 & \text{otherwise.}\end{cases}$$

The conclusion is: for all such $P$ and $Y$, $\mu(M_{P,Y}) \ge 3$.

Edge cases. The lift condition at $t = 0$ requires $M_{P,Y}(0)(\cos s,\sin s)$ to be a positive multiple of $(\cos s, \sin s)$; only angles $s$ admitting such a continuous lift on all of $[0,1]$ contribute to $W$. If $W$ is empty or unbounded, $a = b = 0$ and $\mu = -1$. Only the values of $M_{P,Y}$ on $\tau \in [0,1]$, i.e. of $x$ and $Y$ on $[0,T]$, enter $\mu$. No hypothesis says that $\Sigma$ is nonempty, compact, or that any periodic orbit exists; if $H$ has no periodic orbit the conclusion holds vacuously.