Nondegenerate asymptotic operators with the same Conley–Zehnder index are joined through nondegenerate ones

Let $S_0,S_1:\mathbb R\to\mathbb R^{2n\times2n}$ be continuous $1$-periodic loops of symmetric matrices whose asymptotic operators $-J_0\partial_t-S_0$ and $-J_0\partial_t-S_1$ are nondegenerate, and suppose

Then there is a continuous family $(S_\tau)_{\tau\in[0,1]}$ of continuous $1$-periodic loops of symmetric matrices from $S_0$ to $S_1$ such that every $-J_0\partial_t-S_\tau$ is nondegenerate.

This is the "if" half of the classification of nondegenerate asymptotic operators by their Conley–Zehnder index (Wendl, Theorem 3.34, trivial bundle): each value of the index is attained on a single path component of the space of nondegenerate asymptotic operators.

Formalization Note. $\mathbb R^{2n\times2n}$ is Mat n (real matrices indexed by Fin n ⊕ Fin n) and $J_0=\begin{pmatrix}0&-\mathrm{Id}\\ \mathrm{Id}&0\end{pmatrix}$ is Matrix.J. The family is a continuous map H : C(unitInterval × ℝ, Mat n) (joint continuity in $(\tau,t)\in[0,1]\times\mathbb R$) with H (0, t) = S₀ t, H (1, t) = S₁ t, and, for every $\tau$, IsSymLoop (fun t => H (τ, t)) and IsNondegenerate (fun t => H (τ, t)). Wendl states the theorem for smooth asymptotic operators on a Hermitian bundle with a trivialization; here the bundle is trivial and the loops are continuous.