making ${C^{k,\alpha}}(B,{\mathbb{C}}^m)$ a Banach space (note, however, that $\norm{{\,\cdot\,}}_{k,\alpha}$ must be induced by an appropriate underlying [*adapted norm*]{} [@cabre2003parameterization1 Sec. A.1] on ${\mathbb{R}}^n$ to ensure that $S$ is a contraction). In fact, $S$ is the affine map defined by $$\label{eq:contraction}
        S(\varphi|_B)\coloneqq -P|_B + e^{-A}\left(P|_B + \varphi|_B\right)\circ \Phi^1.$$ Hence we can obtain $\varphi|_B$ by the standard contraction mapping theorem, thereby obtaining a function $\psi|_B\in {C^{k,\alpha}}(B,{\mathbb{C}}^m)$ satisfying $\psi|_B \circ \Phi^1|_B = e^A \psi|_B$. We then extend the domain of $\psi|_B$ using the globalization techniques of [@lan2013linearization; @kvalheim2018global] to obtain a globally defined function $\psi \in {C_{\textnormal{loc}}^{k,\alpha}}(Q,{\mathbb{C}}^m)$ satisfying $\psi \circ \Phi^1 = e^A \psi$. Using an argument of Sternberg [@sternberg1957local Lem. 4] in combination with the uniqueness statement of Theorem \[th:main-thm\], we show that the function $\psi$ satisfies , i.e., $\psi$ is actually a linearizing factor for $\Phi^t$ for *all* $t\in {\mathbb{R}}$. We extend the result to the case that $k = \infty$ using a bootstrapping argument.

Our proof of the existence portion of Theorem \[th:main-thm\], outlined above, was inspired by Sternberg’s proof of his linearization theorem [@sternberg1957local Thms 2, 3, 4] and also has strong similarities with the techniques used to prove the existence of semiconjugacies of the type using the parameterization 