An approximate parametrization of the ergodic partition using time averaged observables

An ergodic set in the state space of a measure-preserving dynamical system is an invariant set on which the system is ergodic. Moreover, it comprises points on statistically identical trajectories, i.e., time averages of any function along any two trajectories in the set are equal. The collection of such sets partitions the state space and is called the ergodic partition. We present a computational algorithm that retrieves a set of coordinates for ergodic sets. Those coordinates can be thought of as generalization of action coordinates from theory of Liouville-integrable systems. Dynamics of the system is embedded into the space of time averages of observables along the trajectories. In this space, the problem is formulated as a dimension-reduction problem, which is handled by the Diffusion Maps algorithm. The algorithm is demonstrated on a 2D map with a mixed state space.