Dynamical convexity of a star-shaped level passes to the orbits of bounded action of $C^\infty$-close star-shaped levels

Let $S=H^{-1}(1)\subset\mathbb{R}^4$ be a strictly star-shaped level whose Hamiltonian flow is dynamically convex (every periodic orbit, prime or multiply covered, has $\mu_{CZ}\ge3$). Let $H_k$ be Hamiltonians with strictly star-shaped levels $H_k^{-1}(1)$ and $H_k\to H$ in $C^\infty_{\mathrm{loc}}$. Then for every action bound $A\in\mathbb{R}$ there is $K$ such that for all $k\ge K$, every periodic orbit $R$ of $X_{H_k}$ on $H_k^{-1}(1)$ with $\mathcal A(R)\le A$ satisfies $\mu_{CZ}(R)\ge3$.

$H_k\to H$ 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 $K$. 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.

No non-degeneracy is assumed, either for $H$ or for the $H_k$. The orbits of large action of $H_k$ are not controlled, so the levels $H_k^{-1}(1)$ need not be dynamically convex. This is the form in which dynamical convexity of the limit is used for the approximating contact forms $\lambda_k$ in Section 3 of Hryniewicz (2014).