in [@mohr2016koopman]. We provide existence and uniqueness results for ${C_{\textnormal{loc}}^{k,\alpha}}$ principal eigenfunctions, and we also show that the (a priori non-unique) “pullback algebra” defined in [@mohr2016koopman] is unique under certain conditions. For the case of periodic orbit attractors, principal eigenfunctions essentially coincide with the notion of isostable coordinates defined in [@wilson2018greater; @monga2019phase], except that the definition in these references involves a limit which might not exist except for the “slowest” isostable coordinate. Our techniques shed light on this issue, and our results imply that this limit does in fact always exist for the “slowest” isostable coordinate if the dynamical system is at least smoother than $C^{1,\alpha}_{\textnormal{loc}}$ with $\alpha > 0$. In fact our results imply—assuming that there is a unique and algebraically simple “slowest” real Floquet multiplier—that any corresponding “slowest” $C^1$ isostable coordinate is always unique modulo scalar multiplication for a $C^1$ dynamical system, without the need for any nonresonance or spectral spread assumptions; furthermore, such a unique isostable coordinate always exists if the dynamics are at least smoother than $C^{1,\alpha}_{\textnormal{loc}}$ with $\alpha > 0$. Similarly, if instead there is a unique and algebraically simple “slowest” complex conjugate pair of Floquet multipliers, then any corresponding “slowest” complex conjugate pair of 