condition of the form $\nu({\,\cdot\,},{\,\cdot\,}) < k + \alpha$.

\[rem:main-thm-1-outline\] Here we sketch the proof of the existence statement of Theorem \[th:main-thm\], which is somewhat more involved than the uniqueness proof. (The existence proof also yields uniqueness, but under stronger assumptions than the uniqueness statement in Theorem \[th:main-thm\].) Since global asymptotic stability of $x_0$ implies that $Q$ is diffeomorphic to ${\mathbb{R}}^n$ [@wilson1967structure Lem 2.1], we may assume that $Q = {\mathbb{R}}^n$ and $x_0 = 0$. For now we consider the case $k < \infty$. First, the $k$-nonresonance assumption implies that we can uniquely solve order by order (in the sense of Taylor polynomials) for $P$ up to order $k$. Once we obtain a polynomial $P$ of sufficiently high order, we derive a fixed point equation for the high-order remainder term $\varphi$, where $\psi = P + \varphi$ is the desired linearizing factor. Given a sufficiently small, positively invariant, compact neighborhood $B$ containing the fixed point, the proof of Lemma \[lem-make-approx-exact\] shows that the spectral spread condition $\nu(e^A,{\mathsf{D}}_{x_0}\Phi^1)< k+\alpha$ implies that the restriction $\varphi|_B$ of the desired high-order term is the fixed point of a map $S\colon {C^{k,\alpha}}(B,{\mathbb{C}}^m)\to {C^{k,\alpha}}(B,{\mathbb{C}}^m)$ which is a contraction with respect to the standard ${C^{k,\alpha}}$ norm $\norm{{\,\cdot\,}}_{k,\alpha}$ 