In a unitary frame, the linearized Reeb flow is the fundamental solution of the asymptotic operator's matrix

Let $H:\mathbb{R}^{2n+2}\to\mathbb{R}$ have a strictly star-shaped level $S=H^{-1}(1)$ and let $Q=(x,T)$ be a periodic Reeb orbit on $S$. Let $Z$ be a unitary frame of $\xi$ along $Q$ (a smooth $1$-periodic family of linear isomorphisms $Z(t):\mathbb{R}^{2n}\to\xi_{x(Tt)}$ with $JZ=ZJ_0$ and $\omega_0(Za,Zb)=\langle J_0a,b\rangle$), and let $S_\infty$ be a loop of $2n\times 2n$ matrices such that the asymptotic operator of $Q$ takes the form $A_Q(Z\eta)=Z\,(-J_0\eta'-S_\infty\eta)\qquad\text{for all smooth $1$-periodic }\eta:\mathbb{R}\to\mathbb{R}^{2n}.$ Let $\Psi$ be the fundamental solution of $S_\infty$ ($\Psi(0)=I$, $\Psi'=J_0S_\infty\Psi$) and $\Phi$ the linearized Reeb flow along $Q$ ($\Phi(0)=\mathrm{id}$, $\Phi'(\tau)=DR(x(\tau))\Phi(\tau)$). Then

So in such a frame the linearized Reeb flow on $\xi$, run at speed $T$, is the fundamental solution $\Psi$ of the matrix form of the asymptotic operator. This identifies the Conley–Zehnder data of $Q$ (from $\Phi$) with the matrix data $\Psi$ used for the operator $-J_0\partial_t-S_\infty$.

Formalization Note The setting is FiniteEnergyPlane on V n with CauchyRiemannIndex.Vec n $=\mathbb{R}^{2n}$ and CauchyRiemannIndex.Mat n. The frame is IsUnitaryXiFrame J Q Z, the asymptotic operator is asympOp J Q, $\Phi$ is IsLinReebFlow Q Φ and $\Psi$ is CauchyRiemannIndex.IsFundamentalSolution Sinf Ψ; matrices act through Matrix.toEuclideanLin. No compatibility hypothesis on J is assumed beyond the frame conditions; no symmetry or continuity of Sinf is assumed.