On the Equivalence of Koopman Eigenfunctions and Commuting Symmetries

The Koopman operator framework offers a way to represent a nonlinear system as a linear one. The key to this simplification lies in the identification of eigenfunctions. While various data-driven algorithms have been developed for this problem, a theoretical characterization of Koopman eigenfunctions from geometric properties of the flow is still missing. This paper provides such a characterization by establishing an equivalence between a set of Koopman eigenfunctions and a set of commuting symmetries -- both assumed to span the tangent spaces at every point on a simply connected open set. Based on this equivalence, we derive an explicit formula for the principal Koopman eigenfunctions and prove its uniform convergence on the region of attraction of a locally asymptotically stable equilibrium point, thereby offering a constructive method for computing Koopman eigenfunctions.