method [@cabre2003parameterization1; @cabre2003parameterization2; @cabre2005parameterization3]. We repeat here an observation of [@cabre2003parameterization1 Sec. 3] and [@cabre2005parameterization3 Rem. 5.5] which is also relevant for numerical computations of linearizing semiconjugacies of the type (such as Koopman eigenfunctions) based on our proof of Theorem \[th:main-thm\]. Consider $P$ satisfying , $B$ and $S$ as in Remark \[rem:main-thm-1-outline\], and an initial guess $\psi_0|_B = P|_B + \varphi_0|_B$ for a local linearizing factor. If $${\textnormal{Lip}(S)}\leq \kappa <  1\qquad \norm{S(\varphi_0|_B)-\varphi_0|_B}\leq \delta,$$ then the standard proof of the contraction mapping theorem implies the estimate $$\label{eq:a-posteriori}
        \norm{\varphi|_B - \varphi_0|_B} \leq \delta/(1-\kappa),$$ where $\varphi|_B$ is such that $\psi|_B = P|_B + \varphi|_B$ is the unique actual local linearizing factor. Thus equation furnishes an upper bound on the distance between the initial guess $\varphi_0|_B$ and the true solution $\varphi|_B$, and can be used for [*a posteriori estimates*]{} in numerical analysis.

Theorem \[th:main-thm\] gave conditions ensuring existence and uniqueness of linearizing factors under spectral spread and nonresonance conditions. Before stating Theorem \[th:main-thm-per\], we state a result on the uniqueness of linearizing factors which does not assume any nonresonance conditions. Proposition \[prop:uniqueness-without-nonresonance\] follows immediately from Lemma \[lem:psi-identically-0\] (used to prove the uniqueness statement of Theorem \[th:main-thm\]) and the fact that $Q$ is diffeomorphic to 