{\mathbb{R}}^m \subset {\mathbb{C}}^m$ is real.

***Existence**.* If furthermore $\Phi\in {C_{\textnormal{loc}}^{k,\alpha}}$ and $\nu(e^{A},{\mathsf{D}}_{x_0} \Phi^1) < k + \alpha$, then such a $\psi$ exists and additionally satisfies $$\label{eq:main-th-3}
    \forall t \in {\mathbb{T}}\colon \psi \circ \Phi^t = e^{tA} \psi.$$ In fact, if $P$ is any “approximate linearizing factor” satisfying ${\mathsf{D}}_{x_0}P = B$ and $$\label{eq:main-th-4}
    P \circ \Phi^1 = e^A P + R$$ with ${\mathsf{D}}_{x_0}^i R = 0$ for all integers $0\leq i < k+\alpha$, then $$\label{eq:main-th-5}
    \psi = \lim_{t\to \infty} e^{-tA} P \circ \Phi^t,$$ in the topology of $C^{k,\alpha}$-uniform convergence on compact subsets of $Q$.

Definitions \[def:nonres\] and \[def:nonres-spread\] are not independent. In particular, if $(X,Y)$ is $(\ell-1)$-nonresonant and $\nu(X,Y) < \ell$, then it follows that $(X,Y)$ is $\infty$-nonresonant. Hence an equivalent statement of Theorem \[th:main-thm\] could be obtained by replacing $k$-nonresonance with $\infty$-resonance everywhere (alternatively, for the existence statement only $(k-1)$-nonresonance need be assumed in the case $\alpha = 0$). We prefer to use the stronger-sounding statement of the theorem above since it makes apparent that the set of matrix pairs $(e^A, {\mathsf{D}}_{x_0}\Phi^1)$ satisfying its hypotheses are *open* in the space of all matrix pairs. Openness for $k < \infty$ is immediate, and openness for $k = \infty$ follows the fact that 