  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 $\R^n$, $n\geq 2$. In this paper we give examples of
real analytic homeomorphisms of $\R^2$ such that the non-synchronizability is
stable in the sense that in a full $C^0$ neighborhood of the given map, no
homeomorphism is synchronizable.
