Non-degenerate approximations fixing an unknotted orbit with $\mathrm{sl}=-1$, whose low-action orbits eventually have index $\ge3$: the orbit eventually binds a transverse disk open book

Let $S=H^{-1}(1)\subset\mathbb{R}^4$ be a strictly star-shaped level and let $\bar P=(\bar x,\bar T)$ be a prime periodic orbit on $S$ that is unknotted with $\operatorname{sl}(\bar P)=-1$. Let $H_k$ be Hamiltonians such that

1. every level $H_k^{-1}(1)$ is strictly star-shaped and every $H_k$ is non-degenerate;
2. $H_k=H$ and $dH_k=dH$ at every point of $\bar x(\mathbb{R})$;
3. $H_k\to H$ in $C^\infty_{\mathrm{loc}}$;
4. for every action bound $A$ there is $K$ such that for $k\ge K$ every periodic orbit $R$ on $H_k^{-1}(1)$ with $\mathcal A(R)\le A$ has $\mu_{CZ}(R)\ge3$.

Then for all large $k$, $\bar P$, viewed as a periodic orbit $P_k$ of $X_{H_k}$ (same parametrization $\bar x$ and period $\bar T$), is the binding of a transverse disk open book on $H_k^{-1}(1)$.

A transverse disk open book on $S=G^{-1}(1)$ with binding $Q=(x,T)$ is a family of maps $\mathrm{page}_\theta:\mathbb{R}^2\to\mathbb{R}^4$, $\theta\in\mathbb{R}$, such that

• $(\theta,u)\mapsto\mathrm{page}_\theta(u)$ 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, injective differential up to the boundary);
• the boundary circle runs along $Q$: $\mathrm{page}_\theta(\cos2\pi t,\sin2\pi t)=x(Tt)$;
• $X_G$ is transverse to every page in the interior: $X_G(\mathrm{page}_\theta(u))\notin\operatorname{im}D\mathrm{page}_\theta(u)$ 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 $(\theta,u)$, $|u|<1$.

This is an adapted open book decomposition (AdaptedDiskOpenBook) without the requirement that the pages be global surfaces of section and without the orientation condition on the pages.

$H_k\to H$ in $C^\infty_{\mathrm{loc}}$ means: for every $m\in\mathbb{N}$ and every compact $K\subset\mathbb{R}^4$, $D^mH_k\to D^mH$ uniformly on $K$. A Hamiltonian $H_k$ is non-degenerate when for every periodic orbit $R=(x,T)$ on $H_k^{-1}(1)$ (prime or not) and its linearized flow $Y$, the linearized return map on $\xi$, written in the global frame $(Z_1,Z_2)$ as the matrix $\varphi(1)$ of linearizedXiPath, does not have $1$ as an eigenvalue. The action of a periodic orbit $R=(x,T)$ is $\mathcal A(R)=\int_0^T\lambda_0(x(t))[\dot x(t)]\,dt$, with $\lambda_0=\tfrac12\sum(q_i\,dp_i-p_i\,dq_i)$ the Liouville form (liouvilleForm). The Conley–Zehnder index $\mu_{CZ}(R)$ is czIndexOfPath of the linearized flow on $\xi$ written in the global frame $(Z_1,Z_2)$ (linearizedXiPath), with its lower semicontinuous extension to degenerate orbits.

The hypotheses are exactly those of bounds_global_section_of_approx_of_eventually_czIndex_ge_three. This statement is the open book part of the conclusion of Hryniewicz's Section 3 for the contact forms $\lambda_k$, $k$ large; it does not assert that the pages are global surfaces of section.