our classification theorem for $C^\infty$ eigenfunctions of $C^\infty$ dynamical systems with a globally attracting hyperbolic fixed point or periodic orbit. Finally, §\[sec:proofs-main-results\] contains the proofs of Theorems \[th:main-thm\] and \[th:main-thm-per\].

Acknowledgements {#acknowledgements .unnumbered}
----------------

The majority of this work was performed while Kvalheim was a postoctoral researcher at the University of Michigan. Both authors were supported by ARO grants W911NF-14-1-0573 and W911NF-17-1-0306 to Revzen. We thank George Haller, David Hong, Igor Mezić, Jeff Moehlis, Corbinian Schlosser, and Dan Wilson for helpful discussions and comments. We owe special gratitude to Alexandre Mauroy and Ryan Mohr for their careful reading of the manuscript and valuable feedback; in particular, one of Mauroy’s comments led to a sharpening of the uniqueness statements of Theorems \[th:main-thm\] and \[th:main-thm-per\] and Propositions \[prop:koopman-cka-fix\] and \[prop:koopman-cka-per\], and Mohr found an error in Definition \[def:nonres\].

Main results {#sec:main-results}
============

Before stating our main results, we give two definitions which are essentially asymmetric versions of some appearing in [@sternberg1957local; @sell1985smooth]. When discussing eigenvalues and eigenvectors of a linear map or matrix in the remainder of the paper, we are always discussing eigenvalues and eigenvectors of its complexification, although we do not always make this explicit.

\[def:nonres\]Let $X \in {\mathbb{C}}^{d\times d}$ 