A} B.$$

***Uniqueness.*** Assume that $(e^{\tau A},{\mathsf{D}}_{x_0} \Phi^\tau|_{E^s_{x_0}})$ is $k$-nonresonant, and assume that either $\nu(e^{\tau A},{\mathsf{D}}_{x_0} \Phi^\tau|_{E^s_{x_0}}) < k + \alpha$ or $\nu(e^{\tau A},{\mathsf{D}}_{x_0} \Phi^\tau|_{E^s_{x_0}}) \leq k$. Then any $\psi \in {C_{\textnormal{loc}}^{k,\alpha}}(Q,{\mathbb{C}}^m)$ satisfying $$\label{eq:main-th-per-2}
        \forall t\in {\mathbb{R}}\colon \psi \circ \Phi^t = e^{t A} \psi, \qquad {\mathsf{D}}_{x_0} \psi|_{E^s_{x_0}} = B$$ is unique, and if $B\colon E^s_{x_0}\to {\mathbb{R}}^m$ and $A\in {\mathbb{R}}^{m\times m}$ are real, then $\psi\colon Q\to {\mathbb{R}}^m \subset {\mathbb{C}}^m$ is real.

***Existence.*** If furthermore $\nu(e^{\tau A},{\mathsf{D}}_{x_0} \Phi^\tau|_{E^s_{x_0}}) < k + \alpha$, then such a unique $\psi$ exists.

Applications {#sec:applications}
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In this section, we give some applications of Theorems \[th:main-thm\] and \[th:main-thm-per\] and Proposition \[prop:uniqueness-without-nonresonance\]. §\[sec:app:stern-floq\] contains results on Sternberg linearizations and Floquet normal forms. §\[sec:app-p-eigs\] gives applications to principal Koopman eigenfunctions and isostable coordinates. §\[sec:app-classify\] contains the classification theorems for $C^\infty$ eigenfunctions of a $C^\infty$ dynamical system.

Sternberg linearizations and Floquet normal forms {#sec:app:stern-floq}
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The following result is an improved statement of Sternberg’s linearization theorem for hyperbolic sinks [@sternberg1957local Thms 2,3,4]; it includes uniqueness of the linearizing conjugacy, it sharpens Sternberg’s $C^k$ linearization result to a ${C_{\textnormal{loc}}^{k,\alpha}}$ linearization result, and the linearization is globally defined on all of $Q$ rather than on some neighborhood of $x_0$. Our technique for 