Non-degenerate approximations of a star-shaped Hamiltonian that agree with it to first order along two given periodic orbits

Let $S=H^{-1}(1)\subset\mathbb{R}^4$ be a strictly star-shaped level and let $P=(x_P,T_P)$, $Q=(x_Q,T_Q)$ be periodic orbits of $X_H$ on $S$. Then there is a sequence of Hamiltonians $H_k:\mathbb{R}^4\to\mathbb{R}$ such that

1. every level $H_k^{-1}(1)$ is strictly star-shaped;
2. every $H_k$ is non-degenerate: no periodic orbit of $X_{H_k}$ on $H_k^{-1}(1)$, prime or multiply covered, has $1$ as a Floquet multiplier of its linearized return map on $\xi$;
3. $H_k=H$ and $dH_k=dH$ at every point of $x_P(\mathbb{R})\cup x_Q(\mathbb{R})$;
4. $H_k\to H$ in $C^\infty_{\mathrm{loc}}$.

$H_k\to H$ 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 $\xi=TS/\mathbb{R}X_{H_k}$, written in the global frame $(Z_1,Z_2)$ as the matrix $\varphi(1)$ of linearizedXiPath, does not have $1$ as an eigenvalue.

By (3), $X_{H_k}=X_H$ along both orbits, so $P$ and $Q$ stay periodic orbits of every $X_{H_k}$. Only the $2$-jet transverse to the orbits is changed, which is what makes $P$ and $Q$ themselves non-degenerate.

In contact language, $H_k^{-1}(1)$ corresponds to the form $g_k\lambda$ on $S^3$ with $g_k\to1$ in $C^\infty$ and $g_k\equiv1$, $dg_k\equiv0$ on the two orbits. Non-degenerate forms are residual in $C^\infty$ (Kupka–Smale for Reeb flows), and the residual set can be chosen inside the forms with a prescribed $1$-jet along finitely many closed orbits. This is the perturbation used in the proof of Lemma 3.12 of Hryniewicz (2014).