Q\to {\mathbb{C}}$ is a Koopman [*eigenfunction*]{} if $\psi$ is not identically zero and satisfies $$\label{eq:koopman-efunc}
    \forall t \in {\mathbb{T}}\colon \psi \circ \Phi^t = e^{\mu t} \psi$$ for some $\mu \in {\mathbb{C}}$. The following are intrinsic definitions of principal eigenfunctions and the principal algebra which extend the definitions for linear systems given in [@mohr2016koopman Def. 2.2–2.3]; a more detailed comparison is given later in Remark \[rem:mohr-mezic\]. The condition $\psi|_M \equiv 0$ was motivated in part by the definition of a certain space ${\mathcal{F}}_{A_c}$ of functions in [@mauroy2016global p. 3358].

\[def:principal-eigenfunction\] Suppose that $\Phi$ has a distinguished, closed, invariant subset $M\subset Q$. We say that an eigenfunction $\psi \in C^1(Q)$ is a [*principal*]{} eigenfunction if $$\label{eq:p-eig-def}
        \psi|_M \equiv 0 \qquad \text{and} \qquad \forall x\in M\colon {\mathsf{D}}_x\psi \neq 0.$$ We define the ${C_{\textnormal{loc}}^{k,\alpha}}$ principal algebra ${\mathcal{A}}^{k,\alpha}_{\Phi}$ to be the complex subalgebra of ${C_{\textnormal{loc}}^{k,\alpha}}(Q,{\mathbb{C}})$ generated by all ${C_{\textnormal{loc}}^{k,\alpha}}$ principal eigenfunctions.

In the case that $\Phi|_{M\times {\mathbb{T}}}$ is minimal (has no proper, closed, nonempty invariant subsets)— can be replaced by the weaker condition $$\label{eq:p-eig-def-weaker}
        \exists x\in M\colon \psi(x) =0 \qquad \text{and} \qquad \exists y\in M\colon {\mathsf{D}}_y\psi \neq 0.$$ This will be the case in the sequel, in which we will consider only the cases that 