Conditioning problems for invariant sets of expanding piecewise affine mappings: Application to loss of ergodicity in globally coupled maps

Abstract

We propose a systematic approach to the construction of invariant union of polytopes (IUP) in expanding piecewise affine mappings. The goal is to characterize ergodic components in these systems. The approach relies on using empirical information embedded in trajectories in order to infer, and then to solve, a so-called conditioning problem for some generating collection of polytopes. A conditioning problem consists of a series of requirements on the polytopes' localisation and on the dynamical transitions between these elements. The core element of the approach is a reformulation of the problem as a set of piecewise linear inequalities for some matrices which encapsulate geometric constraints. In that way, the original topological puzzle is converted into a standard problem in computational geometry. This transformation involves an optimization procedure that ensures that both problems are equivalent, ie. no information is dropped when passing to the analytic formulation. As a proof of concept, the approach is applied to the construction of asymmetric IUP in piecewise expanding globally coupled maps, so that multiple ergodic components result. The resulting mathematical statements explain, complete and extend previous results in the literature, and in particular, they address the dynamics of cluster configurations. Comparison with the numerics reveals that, in all examples, our approach provides sharp existence conditions and accurate fits of the empirical ergodic components