$M$ is either a fixed point or periodic orbit.

Differentiating and using the chain rule immediately yields Propositions \[prop:p-eig-evec-point\] and \[prop:p-eig-evec-cycle\], which have appeared in the literature (see e.g. the proof of [@mauroy2016global Prop. 2]). In these results, $d$ denotes the exterior derivative and ${\mathsf{T}}_{x_0}^* Q$ denotes the cotangent space to $x_0$; $d\psi(x_0)$ corresponds to ${\mathsf{D}}_{x_0}\psi$ after making the canonical identification ${\mathbb{C}}\cong {\mathsf{T}}_{\psi(x_0)}{\mathbb{C}}$.

\[prop:p-eig-evec-point\] Let $x_0$ be a fixed point of the $C^1$ dynamical system $\Phi\colon Q\times {\mathbb{T}}\to Q$. If $\psi$ is a principal Koopman eigenfunction for $\Phi$ satisfying with $\mu\in {\mathbb{C}}$, then for any $t\in {\mathbb{T}}$, it follows that $d \psi(x_0)\in (T_{x_0}^* Q)_{\mathbb{C}}$ is an eigenvector of the (complexified) adjoint $({\mathsf{D}}_{x_0} \Phi^t)^*$ with eigenvalue $e^{\mu t}$.

\[prop:p-eig-evec-cycle\] Let $\Gamma$ be the image of a $\tau$-periodic orbit of the $C^1$ dynamical system $\Phi\colon Q\times {\mathbb{R}}\to Q$. If $\psi$ is a principal Koopman eigenfunction for $\Phi$ satisfying with $\mu\in {\mathbb{C}}$, then for any $x_0 \in \Gamma$, it follows that $d\psi(x_0)\in(T_{x_0}^* Q)_{\mathbb{C}}$ is an eigenvector of the (complexified) adjoint $({\mathsf{D}}_{x_0}\Phi^\tau)^*$ with eigenvalue $e^{\mu \tau}$; in particular, $e^{\mu \tau}$ is a Floquet multiplier for $\Gamma$.

Notice that, for a dynamical system having a globally attracting compact invariant set $M$, any continuous eigenfunction satisfying 