${\mathbb{R}}^{\dim(Q)}$ as mentioned above.

\[prop:uniqueness-without-nonresonance\] Fix $k\in {\mathbb{N}}_{\geq 1}\cup \{\infty\}$ and $0 \leq \alpha \leq 1$, and 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 ${\mathbb{T}}= {\mathbb{R}}$. Let $m\in {\mathbb{N}}_{\geq 1}$ and $e^{A}\in {\mathsf{GL}}(m,
    {\mathbb{C}})$ have spectral radius $\rho(e^{A})<1$ and satisfy either $\nu(e^A,{\mathsf{D}}_0 F) < k + \alpha$ or $\nu(e^A,{\mathsf{D}}_0 F) \leq k$. Let $\varphi\in {C_{\textnormal{loc}}^{k,\alpha}}(Q,{\mathbb{C}}^m)$ satisfy ${\mathsf{D}}_{x_0}^i \varphi = 0$ for all $0\leq i \leq k$ and $$\varphi \circ \Phi^1 = e^{A} \varphi.$$

Then it follows that $\varphi \equiv 0$. In particular, if $\varphi = \psi_1 - \psi_2$, then $$\psi_1 = \psi_2.$$

[Th]{}[ThmMainPer]{}\[th:main-thm-per\] Fix $k\in {\mathbb{N}}_{\geq 1}\cup \{\infty\}$ and $0 \leq \alpha \leq 1$, and let $\Phi\colon Q \times {\mathbb{R}}\to Q$ be a ${C_{\textnormal{loc}}^{k,\alpha}}$ flow having a globally attracting hyperbolic $\tau$-periodic orbit with image $\Gamma\subset Q$, where $Q$ is a smooth manifold. Fix $x_0\in \Gamma$ and let $E^s_{x_0}$ denote the unique ${\mathsf{D}}_{x_0}\Phi^\tau$-invariant subspace complementary to ${\mathsf{T}}_{x_0} \Gamma$. Let $m\in {\mathbb{N}}_{\geq 1}$ and $e^{\tau A}\in {\mathsf{GL}}(m,
        {\mathbb{C}})$ have spectral radius $\rho(e^{\tau A})<1$, and let the linear map $B\colon E^s_{x_0} \to {\mathbb{C}}^m$ satisfy $$\label{eq:main-th-per-1}
        B {\mathsf{D}}_{x_0}\Phi^\tau|_{E^s_{x_0}} = e^{\tau 