## Institute for Mathematical Sciences

## Preprint ims97-3

** P. Le Calvez, M. Martens, C. Tresser, and P. Worfolk**
* Stably Non-synchronizable Maps of the Plane*

Abstract: 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.

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