Stably nonsynchronizable maps of the plane

Pecora and Carroll presented a notion of synchronization where an (n - 1 )-dimensional nonautonomous system is constructed from a given n-dimensional dynamical system by imposing the evolution of one coordinate. They noticed that the resulting dynamics may be contracting even if the original dynamics are not. It is easy to construct flows or maps such that no coordinate has synchronizing properties, but this cannot be done in an open set of linear maps or flows in ℝⁿ, n ≥ 2. In this paper we give examples of real analytic homeomorphisms of ℝ² such that the nonsynchronizability is stable in the sense that in a full C⁰ neighbourhood of the given map, no homeomorphism is synchronizable.