Finding Birkhoff averages via adaptive filtering

In many applications, one is interested in classifying trajectories of symplectic maps as invariant tori, islands, or chaos. Here, we introduce Birkhoff reduced rank extrapolation (RRE), a modified version of the RRE method, which efficiently obtains ergodic averages of invariant tori and islands in symplectic maps. In contrast, Birkhoff RRE does not efficiently converge in chaos, meaning its convergence rate can be used for trajectory classification. Furthermore, for the islands and invariant circles, Birkhoff RRE returns the number of islands and the rotation number. This allows us to find Fourier parameterizations of invariant circles and islands from a single trajectory. We show examples of Birkhoff RRE on the standard map and magnetic field line dynamics.