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 polytopes (IUP) in expanding piecewise affine mappings whose linear components are isotropic scalings. The relies on using empirical information embedded trajectories order infer, and then solve, so-called conditioning problem for some generating collection polytopes. A consists series requirements polytopes’ localisation dynamical transitions between these elements. core element is reformulation as set inequalities matrices which encapsulate geometric constraints. In that way, original topological puzzle converted into standard computational geometry. This transformation involves an optimisation procedure ensures both problems equivalent. As proof concept, applied study loss ergodicity basic examples globally coupled maps. explains, completes substantially extends previous achievements about asymmetric IUP systems. Comparison with numerics reveals sharp existence conditions depending map parameters, accurate fits ergodic components. addition, this application also unanticipated features solutions, especially dependence admissible face directions concerned.