FiniteEnergyPlane.exists_hit_of_tendsto_orbit

Let $H:\mathbb{R}^4\to\mathbb{R}$ have strictly star-shaped level $\{H=1\}$, let $J$ be a compatible $\xi$-structure for $H$, let $P$ be a Reeb orbit of $H$ with "CZ index of $P$ at least $3$", let $u:\mathbb{C}\to\mathbb{R}^4$ be a fast plane having end on $P$ (with sign $+1$), and let $y:\mathbb{R}\to\mathbb{R}^4$ be a Reeb trajectory with $y(0)\notin\mathrm{im}\,P$.

Then both of the following implications hold, where $\mathrm{im}(\mathrm{pr}_H\circ u)=\{\mathrm{pr}_H(u(z)):z\in\mathbb{C}\}$:

1. Forward. If there exist $\tau_0\in\mathbb{R}$ and a real sequence $s_k\to+\infty$ with $y(s_k)\to x_P(\tau_0)$, then for every $n\in\mathbb{R}$ there is $t>n$ with $y(t)\in\mathrm{im}(\mathrm{pr}_H\circ u)$.
2. Backward. If there exist $\tau_0\in\mathbb{R}$ and a real sequence $s_k\to-\infty$ with $y(s_k)\to x_P(\tau_0)$, then for every $n\in\mathbb{R}$ there is $t<n$ with $y(t)\in\mathrm{im}(\mathrm{pr}_H\circ u)$.

$u$ is not assumed embedded, and no base point or energy value of $u$ is prescribed. No nondegeneracy of any orbit and no compactness hypothesis is assumed. The sequence $s_k$ need not be monotone. The conclusion only concerns membership of $y(t)$ in the image; no statement about transversality or the number of hits is made.

Background notions

Conventions. $V_n=\mathbb{R}^{2(n+1)}$ with coordinates $v_0,\dots,v_{2n+1}$; below mostly $n=1$, so $V_1=\mathbb{R}^4$. $Df(p)$ denotes the real Fréchet derivative, which is the zero map by convention where $f$ is not differentiable. For $f$ on $\mathbb{C}$: $\partial_sf(z)=Df(z)[1]$, $\partial_tf(z)=Df(z)[i]$. Also $0^{-1}=0$, and $\mathrm{im}\,P$ is the image of $x_P:\mathbb{R}\to V_n$.

Symplectic data on $V_n$. $(j_0v)_k=-v_{k+1}$ for $k$ even, $(j_0v)_k=v_{k-1}$ for $k$ odd. $\omega_0(u,v)=\sum_k(j_0u)_kv_k=\sum_i(u_{2i}v_{2i+1}-u_{2i+1}v_{2i})$. $\lambda_x(v)=\tfrac12\omega_0(x,v)$. $\nabla H(x)$ has components $DH(x)[e_k]$; $X_H(x)=j_0\nabla H(x)$.

Reeb field. $R_H(x)=\lambda_x(X_H(x))^{-1}\,X_H(x)$ (the zero vector where $\lambda_x(X_H(x))=0$). $\xi$-membership: "$v\in\xi_x$" means $\lambda_x(v)=0$ and $DH(x)[v]=0$.

Strictly star-shaped level. $H\in C^\infty(V_n)$; for every $u\ne0$ there is exactly one $r>0$ with $H(ru)=1$; and $DH(x)[x]>0$ whenever $H(x)=1$. The last clause forces $H(0)\ne1$.

Compatible $\xi$-structure. $J:V_n\to\mathcal{L}(V_n,V_n)$ is $C^\infty$ on all of $V_n$, and for all $x$ with $H(x)=1$ and all $v,w\in\xi_x$: $J_xv\in\xi_x$; $J_xJ_xv=-v$; if $v\ne0$ then $\omega_0(v,J_xv)>0$; $\omega_0(J_xv,J_xw)=\omega_0(v,w)$.

Radial coordinates. $\sigma_H(y)$ is a chosen $r>0$ with $H(ry)=1$ if one exists, and $1$ otherwise. $\mathrm{pr}_H(y)=\sigma_H(y)\,y$ and $a_H(y)=-\tfrac12\log\sigma_H(y)$. (For star-shaped $H$: at $y\ne0$ the $r$ is unique; at $y=0$ the value is $\sigma_H(0)=1$.)

Almost complex structure on $V_n$. For $y\in V_n$ with $x=\mathrm{pr}_H(y)$, $r=\sigma_H(y)$, set $\alpha=\frac{r}{2DH(x)[x]}DH(x)$, $\beta=r\lambda_x$, and $w(h)=h-\alpha(h)\,2y-\beta(h)\,r^{-1}R_H(x)$. Then

$$\tilde J_y(h)=\alpha(h)\,r^{-1}R_H(x)-\beta(h)\,2y+J_x(w(h)).$$

Finite energy curve on $\mathbb{C}$ (only $\mathbb{C}$ itself is used here). $u:\mathbb{C}\to V_n$ with: $u\in C^\infty$; $u(z)\ne0$ for all $z$; $u$ is differentiable at each $z$ with $\partial_su(z)+\tilde J_{u(z)}\partial_tu(z)=0$; and $E(u)<\infty$, where

$$E(u)=\sup_\varphi\int_{\mathbb{C}}\big[\varphi'(a)\big(\partial_sa\,\lambda_w(\partial_tw)-\partial_ta\,\lambda_w(\partial_sw)\big)+\varphi(a)\,\omega_0(\partial_sw,\partial_tw)\big]^+\,dA,$$

with $a=a_H\circ u$, $w=\mathrm{pr}_H\circ u$, the supremum over $C^\infty$ monotone (non-decreasing) $\varphi:\mathbb{R}\to[0,1]$, and $[\cdot]^+$ the positive part (the integral is a $[0,\infty]$-valued Lebesgue integral).

$d\lambda$-energy. For $\Omega\subseteq\mathbb{C}$: $E_{d\lambda}(\Omega,u)=\int_\Omega[\omega_0(\partial_sw,\partial_tw)]^+\,dA\in[0,\infty]$, with $w=\mathrm{pr}_H\circ u$.

Reeb orbit. A Reeb orbit $P$ of $H$ is a pair $(x_P,T_P)$ with $x_P:\mathbb{R}\to V_n$, $T_P>0$, $H(x_P(\tau))=1$, $x_P'(\tau)=R_H(x_P(\tau))$ and $x_P(\tau+T_P)=x_P(\tau)$ for all $\tau$. $T_P$ is any positive period, not necessarily the minimal one, so covers are orbits. $P$ is prime when $x_P(\tau)\ne x_P(0)$ for all $\tau\in(0,T_P)$.

Reeb trajectory. Any $y:\mathbb{R}\to V_n$ such that for every $t\in\mathbb{R}$: $H(y(t))=1$ and $y$ is differentiable at $t$ with $y^\prime(t)=R_H(y(t))$. Nothing excludes $y$ from being periodic or from running along a Reeb orbit.

Nondegenerate orbit. For every $\Phi:\mathbb{R}\to\mathcal{L}(V_n,V_n)$ with $\Phi(0)=\mathrm{id}$ and $\Phi'(\tau)=DR_H(x_Q(\tau))\circ\Phi(\tau)$ for all $\tau$, and every $v\in\xi_{x_Q(0)}$: $\Phi(T_Q)v=v$ implies $v=0$. (Vacuous if no such $\Phi$ exists.)

Nondegenerate end at infinity on $P$ (with sign $+1$; written "$u$ has end on $P$"). There exist reals $d,c$ and $b>0$ such that, as $s\to+\infty$ and uniformly in $t\in\mathbb{R}$,

$$a_H\big(u(e^{2\pi(s+it)})\big)-T_Ps-d\to0,\qquad e^{bs}\big(\mathrm{pr}_H(u(e^{2\pi(s+it)}))-x_P(T_Pt+c)\big)\to0.$$

Fast plane. $u:\mathbb{C}\to V_1$ such that:

1. $u$ is a finite energy curve on $\mathbb{C}$;
2. there exists a prime Reeb orbit $P'$ of $H$ on which $u$ has an end in the above sense;
3. with $w=\mathrm{pr}_H\circ u$ and $\pi_u(z)=\mathrm{c}_H\big(w(z),\ \partial_sw(z)-\lambda_{w(z)}(\partial_sw(z))R_H(w(z))\big)\in\mathbb{C}$, the zero set $Z$ of $\pi_u$ is finite, and there are integers $\mathrm{idx}(z)$, $z\in Z$, summing to $0$, such that for each $z\in Z$ there is $\rho_0>0$ with $\pi_u\ne0$ on $\bar B(z,\rho_0)\setminus\{z\}$ and, for each $\rho\in(0,\rho_0)$, the loop $t\mapsto\pi_u(z+\rho e^{2\pi it})$ has winding number $\mathrm{idx}(z)$.

Here a loop $f:\mathbb{R}\to\mathbb{C}$ "has winding number $k$" when $f$ is continuous, $1$-periodic, nowhere zero, and there is a continuous $\vartheta:\mathbb{R}\to\mathbb{R}$ with $f=|f|e^{i\vartheta}$ and $\vartheta(1)-\vartheta(0)=2\pi k$. An empty $Z$ satisfies clause 3; the definition allows nonempty $Z$ with indices of mixed sign summing to $0$.

The coordinate $\mathrm{c}_H(x,v)\in\mathbb{C}$ is

$$\mathrm{c}_H(x,v)=\omega_0(v,f_2(x))+i\,\omega_0(f_1(x),v),\qquad f_j(x)=|x|^{-1}\Big(Q_j(x)-\frac{DH(x)[Q_j(x)]}{DH(x)[x]}\,x\Big),$$

for $x,v\in\mathbb{R}^4$, with $|x|$ the Euclidean norm, $Q_1(x)=(x_3,x_2,-x_1,-x_0)$ and $Q_2(x)=(-x_2,x_3,x_0,-x_1)$.

Embedded plane. $u:\mathbb{C}\to V_n$ is injective and $Du(z)$ is an injective linear map for every $z$.

Set $\mathcal{E}(H,J,P,K,\gamma)$ of embedded fast planes ($K\subseteq V_1$, $\gamma\in\mathbb{R}$). All $u:\mathbb{C}\to V_1$ that are fast planes, have end on $P$, are embedded planes, have $u(0)\in K$, and satisfy

$$E_{d\lambda}(\{|z|>1\},u)=\max(\gamma,0).$$

When $\gamma\le0$ this says the energy on $\{|z|>1\}$ is $0$. $P$ is not itself required to be prime; the prime orbit $P'$ of the fast plane condition is a separate existential.

$C^\infty_{\mathrm{loc}}$ convergence on $\mathbb{C}$. $v_k\to v$ means: for every compact $K\subseteq\mathbb{C}$ and every $m\in\mathbb{N}$, the $m$-th iterated Fréchet derivatives of $v_k$ converge uniformly on $K$ to that of $v$.

Sequential $C^\infty_{\mathrm{loc}}$ compactness of a set $A$ of maps $\mathbb{C}\to V_1$. Every sequence in $A$ has a subsequence (strictly increasing reindexing) converging in $C^\infty_{\mathrm{loc}}$ on $\mathbb{C}$ to some element of $A$. The empty set has this property.

Hypothesis "CZ index of $Q$ is at least $m$" (for a Reeb orbit $Q$ of $H:V_1\to\mathbb{R}$, $m\in\mathbb{Z}$). Identify $V_1$ with $\mathbb{R}^4$ and let $\mathcal{R}$ be any periodic orbit of $H$ for the Hamiltonian field: $\mathcal{R}=(x_{\mathcal{R}},T_{\mathcal{R}})$, $T_{\mathcal{R}}>0$, $x_{\mathcal{R}}'=X_H(x_{\mathcal{R}})$, $H(x_{\mathcal{R}})=1$, $x_{\mathcal{R}}(t+T_{\mathcal{R}})=x_{\mathcal{R}}(t)$, where $X_H=(-\partial_1H,\partial_0H,-\partial_3H,\partial_2H)$. Then, for every such $\mathcal{R}$ with the same image as $Q$ and with action $T_Q=\int_0^{T_{\mathcal{R}}}\lambda_{x_{\mathcal{R}}}(x_{\mathcal{R}}')\,d\tau$, and for every $Y:\mathbb{R}\to\mathcal{L}(\mathbb{R}^4,\mathbb{R}^4)$ with $Y(0)=\mathrm{id}$, $Y'(t)=DX_H(x_{\mathcal{R}}(t))\circ Y(t)$, one has $m\le\mathrm{cz}(\varphi)$. Here $\varphi:\mathbb{R}\to M_2(\mathbb{R})$ has entries

$$\varphi(\tau)_{ij}=\Big(\mathrm{c}^{\mathbb{R}^2}_H\Big)_i\Big(x_{\mathcal{R}}(T_{\mathcal{R}}\tau),\ \mathrm{pr}^{\xi}\big(Y(T_{\mathcal{R}}\tau)f^{(j)}(x_{\mathcal{R}}(0))\big)\Big),$$

with $\mathrm{c}_H^{\mathbb{R}^2}$ the real pair $(\mathrm{Re}\,\mathrm{c}_H,\mathrm{Im}\,\mathrm{c}_H)$, $f^{(1)}=f_1$, $f^{(2)}=f_2$, and $\mathrm{pr}^\xi(v)=v-\frac{\lambda_x(v)}{\lambda_x(X_H(x))}X_H(x)$ at $x=x_{\mathcal{R}}(T_{\mathcal{R}}\tau)$. The integer $\mathrm{cz}(\varphi)$ is: let $W$ be the set of numbers $(\theta(1)-s)/(2\pi)$ over $s\in[0,2\pi]$ and $\theta$ continuous on $[0,1]$ with $\theta(0)=s$ and $\varphi(t)(\cos s,\sin s)^{\mathsf T}=r(\cos\theta(t),\sin\theta(t))^{\mathsf T}$ for some $r>0$ for each $t\in[0,1]$; let $a=\inf W$, $b=\sup W$ (with the convention that both equal $0$ if $W$ is empty, and each equals $0$ if unbounded on its side). Then $\mathrm{cz}(\varphi)=2(\lceil b\rceil-1)$ if $a\le\lceil b\rceil-1$, and $\mathrm{cz}(\varphi)=2\lceil b\rceil-1$ otherwise. The hypothesis is universally quantified over $\mathcal{R}$ and $Y$ and so is vacuous if no such pair exists.