as linearizing [*factors*]{} or [*factor maps*]{} in the literature and can be viewed as a further generalization of the [*generalized Koopman eigenfunctions*]{} of [@mezic2019spectrum; @korda2019optimal]. We note that such semiconjugacies are distinct from those in the diagram $$\label{eq:cd-param-method}
    \begin{tikzcd}
    &Q \arrow{r}{\Phi^t}&Q\\
    &{\mathbb{C}}^m\arrow{r}{e^{tA}}\arrow{u}{K}&{\mathbb{C}}^m\arrow{u}{K}
    \end{tikzcd}$$ obtained from by flipping the vertical arrows. Existence results for semiconjugacies of the type in were obtained by [@cabre2003parameterization1; @cabre2003parameterization2; @cabre2005parameterization3] in the context of proving invariant manifold results using the parameterization method.

Our main result for the case of a globally attracting hyperbolic fixed point both generalizes and sharpens Sternberg’s linearization theorem [@sternberg1957local Thms 2,3,4] which provides conditions ensuring the existence of a linearizing local $C^k$ diffeomorphism defined on a neighborhood of the fixed point; the results of [@lan2013linearization; @kvalheim2018global] show that this local diffeomorphism can be extended to a global $C^k$ diffeomorphism $\psi \colon Q\to {\mathbb{R}}^n\subset {\mathbb{C}}^n$ making commute. Under Sternberg’s conditions, a corollary of our main result is that this global linearizing diffeomorphism is uniquely determined by its derivative at the fixed point. Additionally, we sharpen Sternberg’s result from $C^k$ to ${C_{\textnormal{loc}}^{k,\alpha}}$ linearizations. For the case of a globally attracting hyperbolic periodic orbit of a flow, our main result also yields a similar existence 