and $\ker({\mathsf{D}}_{x_0}\psi_\theta) = E^s_{x_0}$ are transverse, $\psi$ is an immersion, and $\psi$ is injective since the restriction of $\psi_z$ to any level set ${W^s}_{\Phi^t(x_0)}$ of $\psi_{\theta}$ is the composition of injective maps $e^{tA} \circ \psi_z|_{{W^s}_{x_0}} \circ \Phi^{-t}|_{{W^s}_{\Phi^t(x_0)}}$. $\psi$ is a ${C_{\textnormal{loc}}^{k,\alpha}}$ embedding since, if $(\psi_z|_{{W^s}_{x_0}})^{-1}\colon E^s_{x_0}\to {W^s}_{x_0}$ is the inverse of $\psi_z|_{{W^s}_{x_0}}$, then $\psi^{-1}\colon \psi(Q)\to Q$ $$\label{eq:psi-inv-def}
        \psi^{-1}(\theta,z)\coloneqq  \Phi^{\frac{\arg(\theta)}{2\pi}\tau}\circ (\psi_z|_{{W^s}_{x_0}})^{-1}(  e^{-\frac{\arg(\theta)}{2\pi}\tau  A}z)$$ is the ${C_{\textnormal{loc}}^{k,\alpha}}$ inverse of $\psi$. Since it is clear from that $\psi^{-1}$ is the restriction to $\psi(Q)$ of a continuous function $G\colon S^1\times (E^s_{x_0})_{\mathbb{C}}\to Q$ (given by the same formula), it follows that $\psi$ is a proper map [@lee2011topological Prop. 4.93(e)]. That $\psi^{-1}$ is ${C_{\textnormal{loc}}^{k,\alpha}}$ follows since $(\psi_z|_{{W^s}_{x_0}})^{-1}$ is, and because the definition of $\psi^{-1}$ is independent of the branch of $\arg({\,\cdot\,})$ used since $\Phi^\tau \circ (\psi_z|_{{W^s}_{x_0}})^{-1} \circ e^{- \tau A} = (\psi_z|_{{W^s}_{x_0}})^{-1}$. Hence $\psi$ is a proper ${C_{\textnormal{loc}}^{k,\alpha}}$ embedding, and the same argument shows that $\psi$ is a diffeomorphism onto $S^1\times E^s_{x_0}$ if $A$ is real. This completes the proof.

Principal Koopman eigenfunctions, isostables, and isostable coordinates {#sec:app-p-eigs}
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Given a $C^1$ dynamical system $\Phi\colon Q\times {\mathbb{T}}\to Q$, where $Q$ is a smooth manifold and either ${\mathbb{T}}= {\mathbb{Z}}$ or ${\mathbb{T}}= {\mathbb{R}}$, we say that $\psi \colon 