Relatively Coherent Sets as a Hierarchical Partition Method

Finite time coherent sets [Froyland et al., 2010] have recently been defined by a measure-based objective function describing the degree that sets hold together, along with a Frobenius–Perron transfer operator method to produce optimally coherent sets. Here, we present an extension to generalize the concept to hierarchically define relatively coherent sets based on adjusting the finite time coherent sets to use relative measures restricted to sets which are developed iteratively and hierarchically in a tree of partitions. Several examples help clarify the meaning and expectation of the techniques, as they are the nonautonomous double gyre, the standard map, an idealized stratospheric flow, and empirical data from the Mexico Gulf during the 2010 oil spill. Also for the sake of analysis of computational complexity, we include an Appendix concerning the computational complexity of developing the Ulam–Galerkin matrix estimates of the Frobenius–Perron operator centrally used here.