diffeomorphism.

Theorem \[th:main-thm-per\] implies that a map $\psi_z\in {C_{\textnormal{loc}}^{k,\alpha}}(Q,(E^s_{x_0})_{\mathbb{C}})$ satisfying the conclusions above exists. Letting ${W^s}_{x_0}$ denote the global strong stable manifold (isochron) through $x_0$, we have ${\mathsf{T}}_{x_0}{W^s}_{x_0}= E^s_{x_0}$ and that $\Phi^\tau({W^s}_{x_0}) = {W^s}_{x_0}$. Additionally, ${W^s}_{x_0}$ is a ${C_{\textnormal{loc}}^{k,\alpha}}$ manifold since it is the stable manifold for the fixed point $x_0$ of the ${C_{\textnormal{loc}}^{k,\alpha}}$ diffeomorphism $\Phi^\tau$ [@de1995irwin Thm 2.1]. Proposition \[prop:sternberg\] then implies that $\psi_z|_{{W^s}_{x_0}}\colon {W^s}_{x_0}\to E^s_{x_0} \subset (E^s_{x_0})_{\mathbb{C}}$ is a diffeomorphism.[^2]

Since the vector field generating $\Phi$ intersects ${W^s}_{x_0}$ transversely, a standard argument [@hirsch1974differential p. 243] together with the ${C_{\textnormal{loc}}^{k,\alpha}}$ implicit function theorem [@eldering2013normally Cor. A.4] imply that a real-valued ${C_{\textnormal{loc}}^{k,\alpha}}$ “time-to-impact ${W^s}_{x_0}$” function can be defined on a neighborhood of any point. Using these facts, one can show that the function $\psi_\theta\colon Q\to S^1$ defined via $\psi_\theta({W^s}_{x_0})\equiv 1$ and $\psi_\theta(\Phi^t({W^s}_{x_0})) \equiv e^{2\pi i \frac{t}{\tau}}$ is a ${C_{\textnormal{loc}}^{k,\alpha}}$ function. This function $\psi_\theta$ clearly satisfies $\psi_\theta(x_0) = 1$ and . $\psi_\theta$ is unique among all continuous functions satisfying these equalities, since if $\tilde{\psi}_\theta$ is any other such function, then asymptotic stability of $\Gamma$ implies that the quotient $(\psi_\theta/\tilde{\psi}_\theta)$ is constant on $Q$, and since $(\psi_\theta/\tilde{\psi}_\theta)(x_0) = 1$ it follows that $\tilde{\psi}_\theta \equiv \psi_\theta$.

Since the kernels $\ker ({\mathsf{D}}_{x_0}\psi_z) = {\mathsf{T}}_{x_0}\Gamma$ 