Read-back: FiniteEnergyPlane.exists_not_isCZAtLeast_three_of_windInf_le_one

We work in $\mathbb{R}^4$, with coordinates $0,1,2,3$ and the maximum norm. The theorem takes the following data and hypotheses.

• $H:\mathbb{R}^4\to\mathbb{R}$ has a strictly star-shaped level.
• $J$ is a compatible $\xi$-structure.
• $\Gamma\subset\mathbb{C}$ is a nonempty finite set.
• $u:\mathbb{C}\to\mathbb{R}^4$ is a finite-energy curve on $\mathbb{C}\setminus\Gamma$.
• $P_\infty$ is a Reeb orbit with $\mathrm{NdEnd}(u,e_\infty,P_\infty,+1)$, where $e_\infty(s,t)=e^{2\pi s}e^{2\pi it}$. Explicitly, there are $d,c\in\mathbb{R}$ and $b>0$ with, as $s\to+\infty$ and uniformly in $t$,

$$a(u(e_\infty(s,t)))-T_{P_\infty}s-d\to0,\qquad e^{bs}\big(\mathrm{proj}(u(e_\infty(s,t)))-x_{P_\infty}(T_{P_\infty}t+c)\big)\to0.$$

No nondegeneracy of $P_\infty$ is assumed.

• For every $z\in\Gamma$ there is a nondegenerate Reeb orbit $P_z$ with $\mathrm{NdEnd}(u,e_z,P_z,-1)$, where $e_z(s,t)=z+e^{-2\pi s}e^{-2\pi it}$. Explicitly, for some $d,c$ and $b>0$,

$$a(u(e_z(s,t)))+T_{P_z}s-d\to0,\qquad e^{bs}\big(\mathrm{proj}(u(e_z(s,t)))-x_{P_z}(-T_{P_z}t+c)\big)\to0$$

uniformly in $t$ as $s\to+\infty$.

• $E_{d\lambda}(\mathbb{C}\setminus\Gamma,u)\ne0$.
• $k\in\mathbb{Z}$ with $k\le1$, and there is $s_0$ such that for all $s\ge s_0$ the loop

$$t\mapsto\xi_{\mathbb{C}}\Big(W(s,t),\ \partial_sW(s,t)-\lambda_{W(s,t)}(\partial_sW(s,t))R_H(W(s,t))\Big)$$

has winding number $k$. Here $W(s,t)=\mathrm{proj}(u(e^{2\pi s}e^{2\pi it}))$, and $\partial_sW$ is the Fréchet derivative of $W:\mathbb{R}^2\to\mathbb{R}^4$ applied to $(1,0)$ ($0$ where $W$ is not differentiable).

Conclusion. There exist $z\in\Gamma$ and a Reeb orbit $P_z$ with the following properties:

• $P_z$ is nondegenerate;
• $\mathrm{NdEnd}(u,e_z,P_z,-1)$ holds;
• $P_z.\mathrm{IsCZAtLeast}(3)$ fails.

The failure means there exist a periodic orbit $(y,T_y)$ of the Hamiltonian system of $H$ and a linearized flow $Y$ along it with the following properties (all terms are defined below):

• $y$ has the same image as $x_{P_z}$;
• the action satisfies $T_{P_z}=\int_0^{T_y}\lambda_{y}(y')\,d\tau$;
• $\mu_{\mathrm{CZ}}(\Psi_{y,Y})\le2$, that is, $\mu_{\mathrm{CZ}}(\Psi_{y,Y})<3$.

The orbit $P_z$ in the conclusion need not be the one supplied by the hypothesis for that $z$.

Definitions used

Forms.

• $j_0(v)=(-v_1,v_0,-v_3,v_2)$.
• $\omega_0(u,v)=u_0v_1-u_1v_0+u_2v_3-u_3v_2$.
• $\lambda_x(v)=\tfrac12\omega_0(x,v)$.
• $X_H=j_0\nabla H=(-\partial_1H,\partial_0H,-\partial_3H,\partial_2H)$, with $\nabla H(x)_k=DH(x)e_k$.
• $R_H(x)=\lambda_x(X_H(x))^{-1}X_H(x)$, with $0^{-1}=0$.
• $\xi_x=\{v:\lambda_x(v)=0,\ DH(x)v=0\}$.

Strictly star-shaped level. $H$ is $C^\infty$; every $u\ne0$ has a unique $r>0$ with $H(ru)=1$; and $DH(x)x>0$ on $\{H=1\}$.

Compatible $\xi$-structure. $J:\mathbb{R}^4\to L(\mathbb{R}^4)$ is $C^\infty$, and for $H(x)=1$ and $v,w\in\xi_x$:

• $J(x)v\in\xi_x$;
• $J(x)^2v=-v$;
• $\omega_0(v,J(x)v)>0$ for $v\ne0$;
• $\omega_0(J(x)v,J(x)w)=\omega_0(v,w)$.

Reeb orbit. $Q=(x_Q,T_Q)$ with $T_Q>0$, $H(x_Q)=1$, $x_Q'=R_H(x_Q)$, and $x_Q(\tau+T_Q)=x_Q(\tau)$. The period $T_Q$ is not necessarily minimal.

Nondegenerate. For every $\Phi$ with $\Phi(0)=\mathrm{id}$ and $\Phi'(\tau)=DR_H(x_Q(\tau))\Phi(\tau)$, the only $v\in\xi_{x_Q(0)}$ with $\Phi(T_Q)v=v$ is $0$.

Radial coordinates. $r(y)$ is a chosen $r>0$ with $H(ry)=1$ if one exists, and $r(y)=1$ otherwise. Then $\mathrm{proj}(y)=r(y)y$ and $a(y)=-\tfrac12\log r(y)$.

$\tilde J$. With $x=\mathrm{proj}(y)$, $r=r(y)$, $\alpha(v)=\frac{r}{2DH(x)x}DH(x)v$ and $\beta(v)=r\lambda_x(v)$:

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

Energies. Write $a_u=a\circ u$ and $w_u=\mathrm{proj}\circ u$, with $\partial_sf=Df(z)1$ and $\partial_tf=Df(z)i$.

• $E_{d\lambda}(U,u)=\int_U\max(\omega_0(\partial_sw_u,\partial_tw_u),0)\,dz$.
• $E(U,u)$ is the supremum, over $C^\infty$ nondecreasing $\phi:\mathbb{R}\to[0,1]$, of

$$\int_U\max\big(\phi'(a_u)(\partial_sa_u\lambda_{w_u}(\partial_tw_u)-\partial_ta_u\lambda_{w_u}(\partial_sw_u))+\phi(a_u)\omega_0(\partial_sw_u,\partial_tw_u),0\big)\,dz.$$

Finite-energy curve on $U$. This means all of the following:

• $u$ is $C^\infty$ on $U$ and nonvanishing there;
• $u$ is differentiable with $\partial_su+\tilde J(u)\partial_tu=0$ at each $z\in U$;
• $E(U,u)<\infty$.

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

• $f$ is continuous, $1$-periodic and nowhere zero;
• some continuous $\theta$ has $f=|f|e^{i\theta}$;
• $\theta(1)-\theta(0)=2\pi k$.

$\xi$-coordinates. $\xi_{\mathbb{C}}(x,v)=\omega_0(v,e_2(x))+i\,\omega_0(e_1(x),v)$, where

$$e_1(x)=|x|_2^{-1}\Big(q_2(x)-\tfrac{DH(x)q_2(x)}{DH(x)x}x\Big),\qquad e_2(x)=|x|_2^{-1}\Big(q_1(x)-\tfrac{DH(x)q_1(x)}{DH(x)x}x\Big).$$

Here $q_1(x)=(-x_2,x_3,x_0,-x_1)$, $q_2(x)=(x_3,x_2,-x_1,-x_0)$, and $|\cdot|_2$ is the Euclidean norm.

$Q.\mathrm{IsCZAtLeast}(3)$. This condition concerns every periodic orbit $(y,T_y)$ of the Hamiltonian system that satisfies the following:

• $T_y>0$, $y'=X_H(y)$, $H(y)=1$ and $y(t+T_y)=y(t)$;
• $\{y(\tau)\}=\{x_Q(\tau)\}$ and $T_Q=\int_0^{T_y}\lambda_{y(\tau)}(y'(\tau))\,d\tau$.

For each such orbit and every $Y$ with $Y(0)=\mathrm{id}$ and $Y'(t)=DX_H(y(t))Y(t)$, it requires $3\le\mu_{\mathrm{CZ}}(\Psi_{y,Y})$. The ingredients are defined as follows.

• $\Psi_{y,Y}(\tau)\in\mathbb{R}^{2\times2}$ has $(i,j)$ entry equal to the $i$-th real component of $\xi_{\mathbb{C}}$ at $y(T_y\tau)$, applied to $v-\frac{\lambda(v)}{\lambda(X_H)}X_H$ at $y(T_y\tau)$, where $v=Y(T_y\tau)e_{j+1}(y(0))$.
• $W_\Psi$ is the set of numbers $(\theta(1)-s)/2\pi$ over all $s\in[0,2\pi]$ and all $\theta$ continuous on $[0,1]$ with $\theta(0)=s$ such that $\Psi(t)(\cos s,\sin s)$ is a positive multiple of $(\cos\theta(t),\sin\theta(t))$ for all $t\in[0,1]$.
• With $a=\inf W_\Psi$ and $b=\sup W_\Psi$ (real infimum and supremum, with junk values if the set is empty or unbounded), $\mu_{\mathrm{CZ}}=2(\lceil b\rceil-1)$ if $a\le\lceil b\rceil-1$, and $2\lceil b\rceil-1$ otherwise.