isostable coordinates are always unique modulo scalar multiplication for a $C^1$ dynamical system, and such a unique pair always exists if the dynamics are $C^{1,\alpha}_{\textnormal{loc}}$ with $\alpha > 0$. As a final application of our main results, we give a complete classification of $C^\infty$ eigenfunctions for a $C^\infty$ dynamical system with semisimple (diagonalizable over ${\mathbb{C}}$) and nonresonant linearization, generalizing known results for analytic dynamics and analytic eigenfunctions [@mauroy2013isostables; @mezic2019spectrum].

The remainder of the paper is organized as follows. In §\[sec:main-results\] we state Theorems \[th:main-thm\] and \[th:main-thm-per\], our two main results, without proof. We also state a proposition on the uniqueness of linearizing factors which does not assume any nonresonance conditions. As applications we derive in §\[sec:applications\] several results which are essentially corollaries of this proposition and the two main theorems. §\[sec:app:stern-floq\] contains existence and uniqueness theorems for global Sternberg linearizations and Floquet normal forms. In §\[sec:app-p-eigs\] we define principal Koopman eigenfunctions and isostable coordinates for nonlinear systems and show how Theorems \[th:main-thm\] and \[th:main-thm-per\] yield corresponding existence and uniqueness results. We then discuss the relationship between various notions defined in [@mohr2016koopman] and our definitions, and we also discuss the convergence of the isostable coordinate limits in [@wilson2018greater; @monga2019phase]. §\[sec:app-classify\] contains 