On the numerical approximation of the Perron-Frobenius and Koopman operator

Information about the behavior of dynamical systems can oftenbe obtained by analyzing the eigenvalues and corresponding eigenfunctions oflinear operators associated with a dynamical system. Examples of such operatorsare the Perron-Frobenius and the Koopman operator. In this paper, wewill review di erent methods that have been developed over the last decades tocompute nite-dimensional approximations of these in nite-dimensional operators- in particular Ulam's method and Extended Dynamic Mode Decomposition(EDMD) - and highlight the similarities and di erences between theseapproaches. The results will be illustrated using simple stochastic di erentialequations and molecular dynamics examples.