Statistically optimal almost-invariant sets

Abstract

Chaotic dynamical systems are often transitive, although this transitivity is sometimes very weak. It is of interest to divide the phase space into large regions, between which there is relatively little communication of trajectories. We present fast, simple algorithms to find such divisions. The present work builds on the results of Froyland and Dellnitz [G. Froyland, M. Dellnitz, Detecting and locating near-optimal almost-invariant sets and cycles, SIAM J. Sci. Comput. 24 (6) (2003) 1839–1863], focussing on a statistical description of transitivity that takes into account the fact that trajectories tend to visit different regions of phase space with different frequencies. The new work takes advantage of theoretical results from the theory of reversible Markov chains. A new adaptive algorithm is put forward to efficiently deal with situations where the boundaries of the weakly communicating regions are complicated. This algorithm is illustrated with the standard map. Relevant convergence results are proven.