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.