globalizing the domain of the linearization is essentially the same as those used in [@lan2013linearization; @kvalheim2018global].

\[prop:sternberg\] Fix $k\in {\mathbb{N}}_{\geq 1}\cup \{\infty\}$ and $0 \leq \alpha \leq 1$. Let $\Phi\colon Q \times {\mathbb{T}}\to Q$ be a ${C_{\textnormal{loc}}^{k,\alpha}}$ dynamical system having a globally attracting hyperbolic fixed point $x_0\in Q$, where $Q$ is a smooth manifold and either ${\mathbb{T}}= {\mathbb{Z}}$ or ${\mathbb{T}}= {\mathbb{R}}$. Assume that $\nu({\mathsf{D}}_{x_0} \Phi^1,{\mathsf{D}}_{x_0} \Phi^1) < k + \alpha$, and assume that $({\mathsf{D}}_{x_0} \Phi^1,{\mathsf{D}}_{x_0} \Phi^1)$ is $k$-nonresonant.

Then there exists a unique diffeomorphism $\psi\in {C_{\textnormal{loc}}^{k,\alpha}}(Q,{\mathsf{T}}_{x_0}Q)$ satisfying $$\label{eq:sternberg-lin}
        \forall t \in {\mathbb{T}}\colon \psi \circ \Phi^t = {\mathsf{D}}_{x_0} \Phi^t \psi, \qquad {\mathsf{D}}_{x_0}\psi = {\textnormal{id}}_{{\mathsf{T}}_{x_0}Q}.$$ (In writing ${\mathsf{D}}_{x_0}\psi = {\textnormal{id}}_{{\mathsf{T}}_{x_0}Q}$, we are making the standard and canonical identification ${\mathsf{T}}_{0}({\mathsf{T}}_{x_0}Q)\cong {\mathsf{T}}_{x_0} Q$.)

Identifying ${\mathsf{T}}_{x_0} Q$ with ${\mathbb{R}}^n$ by choosing a basis, we apply Theorem \[th:main-thm\] with $e^{tA} = {\mathsf{D}}_{x_0} \Phi^t$ and $B = {\textnormal{id}}_{{\mathsf{T}}_{x_0} Q}$ to obtain a unique $\psi\in {C_{\textnormal{loc}}^{k,\alpha}}(Q,{\mathsf{T}}_{x_0} Q)$ satisfying and ${\mathsf{D}}_{x_0} \psi = {\textnormal{id}}_{{\mathsf{T}}_{x_0}Q}$. It remains only to show that $\psi$ is a diffeomorphism. To do this, we separately show that $\psi$ is injective, surjective, and a local diffeomorphism.

By continuity, ${\mathsf{D}}_{x_0} \psi = {\textnormal{id}}_{{\mathsf{T}}_{x_0}Q}$ implies that ${\mathsf{D}}_x \psi$ is invertible for all $x$ in some neighborhood $U \ni 