The analysis of dynamical systems typically taught in introductory graduate courses focuses on analysis of trajectories: curves in the state space, parametrized by time. While the trajectory-based approach is successful in analyzing exceptional solutions, e.g., stable and unstable manifolds attached to fixed points, it is a poor choice for recognizing coarse patterns in state spaces, e.g., vortices in fluid-like flows.

An alternative is the operator-theoretic description of dynamical systems. This doctoral project studied the Koopman operator, which describes how functions, or observables, change as the dynamical system evolves in time. The main interest was in identification of invariant sets in the state space, where behavior of the system is uniform ''on average''. The arrangement of invariant sets is connected to the ergodic quotient: a subset of a sequence space where trajectories are described using values of invariant functions along them. By endowing the ergodic quotient with a Sobolev space metric structure, we are able to identify coherent sets: invariant subsets in the state space that possess a complete set of continuous invariant functions. These coherent sets correspond to continuous segments in the ergodic quotient.

The ergodic quotient analysis is model-free: it requires only data obtained by simulation or through experiments. As a proof-of-concept, we implemented the algorithm as a numerical code that approximates the ergodic quotient using trajectory averages of a function basis on the state space. The geometry of the ergodic quotient is analyzed with help of a machine-learning algorithm, Diffusion Maps, which enables computational identification of continuous segments and, consequently, partitioning of the state space into coherent sets.

To demonstrate effectiveness, several examples of dynamical systems were analyzed, ranging from iterated maps with mixed state spaces to forced fluid-like flows. The results obtained by the ergodic quotient analysis match the results of previous analyses of the systems. To conclude, further research directions are proposed, based on the results of the doctoral project.