Properties of Koopman operator and its approximations for dynamical systems with symmetries.*

We investigate the properties of Koopman operator governing the evolution of observables for dynamical systems with point symmetries. Koopman operator approximations such as extended dynamic mode decomposition (EDMD) are a powerful tool for studying nonlinear dynamical systems with unknown underlying dynamics directly from data. We consider the properties of the EDMD approximation of Koopman operator for such systems and propose a modification of the algorithm that utilizes the properties of the isotypic decomposition of dynamical systems with symmetries.