and $Y \in {\mathbb{C}}^{n\times n}$ be matrices with eigenvalues $\mu_1,\ldots,\mu_d$ and $\lambda_1,\ldots,\lambda_n$, respectively, repeated with multiplicities. For any $k \in {\mathbb{N}}_{\geq 1}$, we say that $(X,Y)$ is [*$k$-nonresonant*]{} if, for any $i\in \{1,\ldots, d\}$ and any $m=(m_1,\ldots, m_n) \in {\mathbb{N}}^n_{\geq 0}$ satisfying $2\leq m_1 + \cdots + m_n \leq  k$, $$\label{eq:nonres-lem}
        \mu_i \neq \lambda_1^{m_1}\cdots \lambda_n^{m_n}.$$ (Note this condition vacuously holds if $k = 1$.) We say $(X,Y)$ is [*$\infty$-nonresonant*]{} if $(X,Y)$ is $k$-nonresonant for every $k\in {\mathbb{N}}_{\geq 1}$.

For the definition below, recall that the spectral radius $\rho(X)$ of a matrix is defined to be the largest modulus (absolute value) of the eigenvalues of (the complexification of) $X$.

\[def:nonres-spread\] Let $X\in {\mathsf{GL}}(m,{\mathbb{C}})$ and $Y\in {\mathsf{GL}}(n,{\mathbb{C}})$ be invertible matrices with the spectral radius $\rho(Y)$ satisfying $\rho(Y) < 1$. We define the spectral spread $\nu(X,Y)$ to be $$\label{eq:spread-def}
        \nu(X,Y)\coloneqq \max_{\substack{\mu \in \textnormal{spec}(X)\\ \lambda \in \textnormal{spec}(Y)}}\frac{\ln(|\mu|)}{\ln(|\lambda|)}.$$

Finally, here we recall the definition of ${C_{\textnormal{loc}}}^{k,\alpha}$ functions.

\[def:ckal-funcs\] Let $M,N$ smooth manifolds of dimensions $m$ and $n$, let $\psi\in C^k(M,N)$ a $C^k$ map $\psi\colon M\to N$ with $k\in {\mathbb{N}}_{\geq 0}$, and let $0\leq \alpha \leq 1$. We will say that $\psi \in {C_{\textnormal{loc}}}^{k,\alpha}(M,N)$ if for every $x\in M$ there exist charts $(U_1,\varphi_1)$ and $(U_2,\varphi_2)$ containing 