Non-degenerate approximations fixing an unknotted orbit with $\mathrm{sl}=-1$, whose low-action orbits eventually have index $\ge3$: the orbit eventually bounds a disk-like global section

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 of $X_{H_k}$ (any $P_k$ with the same parametrization $\bar x$ and period $\bar T$), bounds a disk-like global surface of section for the flow on $H_k^{-1}(1)$.

$H_k\to H$ 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.

Hypothesis 4 controls only orbits of bounded action, so the levels $H_k^{-1}(1)$ are not assumed dynamically convex, and the limit level $S$ is not assumed dynamically convex or non-degenerate. In Hryniewicz's notation $H_k^{-1}(1)$ carries the non-degenerate contact form $\lambda_k=h_k\lambda$ with $h_k\to1$, $h_k\equiv1$, $dh_k\equiv0$ on $\bar x(\mathbb{R})$ (his (29)). This is the step of his Section 3 that produces disk-like global sections for $\lambda_k$ when $k$ is large.