A trajectory whose page parameter stays bounded on a half-line converges to the binding

Let $y$ be a trajectory of $X_H$ on $S$ with a page parameter $(\Theta,u)$, $\Theta$ continuous and strictly monotone. If $\Theta$ is bounded on $[a,\infty)$ for some $a$, then $\operatorname{dist}(y(t),x(\mathbb{R}))\to0$ as $t\to+\infty$. If $\Theta$ is bounded on $(-\infty,a]$ for some $a$, then $\operatorname{dist}(y(t),x(\mathbb{R}))\to0$ as $t\to-\infty$.

Here $S=H^{-1}(1)\subset\mathbb{R}^4$ is a strictly star-shaped level, $P=(x,T)$ is a periodic orbit on $S$, and $(\mathrm{page}_\theta)_{\theta\in\mathbb{R}}$ is a transverse disk open book on $S$ with binding $P$:

• $(\theta,u)\mapsto\mathrm{page}_\theta(u)$ is $C^\infty$ and $1$-periodic in $\theta$;
• each $\mathrm{page}_\theta$ restricts to a smooth embedding of the closed unit disk $\bar{\mathbb D}$ into $S$ (IsSmoothDiskEmbedding);
• the boundary circle runs along $P$: $\mathrm{page}_\theta(\cos2\pi t,\sin2\pi t)=x(Tt)$;
• $X_H$ is transverse to every page in the interior: $X_H(\mathrm{page}_\theta(u))\notin\operatorname{im}D\mathrm{page}_\theta(u)$ for $|u|<1$;
• $(\theta,u)\mapsto\mathrm{page}_\theta(u)$ maps $[0,1)\times\mathbb D$ bijectively onto $S\setminus x(\mathbb{R})$, with injective differential at every $(\theta,u)$, $|u|<1$.

These are exactly the open book hypotheses of isDiskLikeGlobalSectionMap_of_transverse_open_book.

A page parameter along a trajectory $y$ of $X_H$ on $S$ is a pair of functions $\Theta:\mathbb{R}\to\mathbb{R}$, $u:\mathbb{R}\to\mathbb D$ (open unit disk) with $y(t)=\mathrm{page}_{\Theta(t)}(u(t))$ for all $t$. Distances to the binding are $\operatorname{dist}(y(t),x(\mathbb{R}))$ (Metric.infDist, for the sup norm on $\mathbb{R}^4$; any norm gives the same convergence).

Role: a trajectory that crosses only finitely many pages in forward (or backward) time accumulates only on the binding. No index hypothesis on $P$ is needed.