Stably nonsynchronizable maps of the plane

Pecora and Carroll presented a notion of synchronization where an (n - I)-d imensional nonautonomous system is constructed from a given n-dimensional d ynamical system by imposing the evolution of one coordinate. They noticed t hat 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 Bows in R-n, n greater than or equal to 2. In this paper we give e xamples of real analytic homeomorphisms of R-2 such that the nonsynchroniza bility is stable in the sense that in a full CO neighbourhood of the given map, no homeomorphism is synchronizable.