with $\mu \in {\mathbb{C}}$ must have $|e^{\mu}|\leq 1$. If this attracting set $M$ is furthermore a hyperbolic fixed point, then there is the stronger conclusion that either $\mu = 0$ or $|e^{\mu}|< 1$. With this observation, Proposition \[prop:koopman-cka-fix\] below is now immediate from Theorem \[th:main-thm\] and Proposition \[prop:p-eig-evec-point\]. We remind the reader of Remark \[rem:point-cinf-case\] which points out that, in the case $k = \infty$, no spectral spread conditions are needed because all inequalities of the form $\nu({\,\cdot\,},{\,\cdot\,}) < \infty$ trivially hold.

\[prop:koopman-cka-fix\] 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 either ${\mathbb{T}}= {\mathbb{Z}}$ or ${\mathbb{T}}= {\mathbb{R}}$. Fix $k \in {\mathbb{N}}_{\geq 1}\cup\{+\infty\}$ and $0 \leq \alpha \leq 1$, fix $\mu \in {\mathbb{C}}$, and let $\psi_1\in {C_{\textnormal{loc}}^{k,\alpha}}(Q,{\mathbb{C}})$ be any Koopman eigenfunction satisfying with $\mu \in {\mathbb{C}}$.

***Uniqueness of Koopman eigenvalues and principal eigenfunctions.*** Assume that either\
$\nu(e^{\mu},{\mathsf{D}}_{x_0}\Phi^1) < k + \alpha$ or $\nu(e^{\mu},{\mathsf{D}}_{x_0}\Phi^1) \leq k$.

1.  \[item:uniq-thm-1\] Then there exists $m = (m_1,\ldots,m_n)\in {\mathbb{N}}_{\geq 0}^n$ such that $$e^{\mu} =  e^{m\cdot \lambda},$$ where $e^{\lambda_1},\ldots,e^{\lambda_n}$ are the eigenvalues of ${\mathsf{D}}_{x_0} \Phi^1$ repeated with multiplicities and $\lambda\coloneqq (\lambda_1,\ldots,\lambda_n)$.

2.  \[item:uniq-thm-2\] Additionally assume that $\psi_1$ is a principal eigenfunction so that $e^{\mu} 