be identified with the linear map ${\mathsf{D}}_{x}^i F\colon ({\mathsf{T}}_{x}M)^{\otimes i} \to {\mathsf{T}}_{F(x)}N$ from the $i$-th tensor power $({\mathsf{T}}_{x}M)^{\otimes i}$ to ${\mathsf{T}}_{x}N$ represented in local coordinates by the $(1+i)$-dimensional array of $i$-th partial derivatives of $F$ evaluated at $x$.

We now state our main results, Theorems \[th:main-thm\] and \[th:main-thm-per\], as well as Proposition \[prop:uniqueness-without-nonresonance\]. Figure \[def:nonres-spread\] illustrates the condition $\nu(e^{A},{\mathsf{D}}_{x_0} \Phi^1) < k + \alpha$ in Theorem \[th:main-thm\] below.

[Th]{}[ThmMain]{}\[th:main-thm\] Let $\Phi\colon Q \times {\mathbb{T}}\to Q$ be a $C^{1}$ 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 ${\mathbb{T}}= {\mathbb{R}}$. Let $m\in {\mathbb{N}}_{\geq 1}$ and $e^{A}\in {\mathsf{GL}}(m,
    {\mathbb{C}})$ have spectral radius $\rho(e^{A})<1$, and let the linear map $B\colon {\mathsf{T}}_{x_0}Q\to {\mathbb{C}}^m$ satisfy $$\label{eq:main-th-1}
    \forall t \in {\mathbb{T}}\colon B {\mathsf{D}}_{x_0}\Phi^t = e^{tA} B.$$

***Uniqueness.*** Fix $k\in {\mathbb{N}}_{\geq 1}\cup \{\infty\}$ and $0 \leq \alpha \leq 1$, assume that $(e^{A},{\mathsf{D}}_{x_0} \Phi^1)$ is $k$-nonresonant, and assume that either $\nu(e^{A},{\mathsf{D}}_{x_0} \Phi^1) < k + \alpha$ or $\nu(e^{A},{\mathsf{D}}_{x_0} \Phi^1) \leq k$. Then any $\psi \in {C_{\textnormal{loc}}^{k,\alpha}}(Q,{\mathbb{C}}^m)$ satisfying $$    \psi \circ \Phi^1 = e^{A} \psi, \qquad {\mathsf{D}}_{x_0} \psi = B$$ is unique, and if $B\colon {\mathsf{T}}_{x_0}Q\to {\mathbb{R}}^m$ and $A\in {\mathbb{R}}^{m\times m}$ are real, then $\psi\colon Q\to 