Stability of Fixed Points for Nonlinear Selfconsistent Transfer Operators via Cone Contractions

Abstract

In this paper we investigate the action of self-consistent transfer operators (STOs) on Birkhoff cones and give sufficient conditions for stability of their fixed points. Our approach relies on the order preservation properties of STOs that can be established via the study of their differential. We show that this approach is effective both in the weak coupling regime and in some strong coupling ones. In particular, we apply the construction to STOs arising from strongly coupled maps both deterministic and noisy. Our approach allows for explicit estimates that we use to give examples of STOs with multiple stable fixed points. Furthermore we show examples where some of these fixed points are far from the asymptotic statistical behaviour of the corresponding system of finite coupled maps