We study the dynamics of planar diffeomorphisms having a unique fixed point that is a hyperbolic local saddle. We obtain sufficient conditions under which the fixed point is a global saddle. We also address the special case of D2-symmetric maps, for which we obtain a similar result for C1 homeomorphisms. Some applications to differential equations are also given.

This is joint work with B. Alarc ́on (UFF — Brasil) and S.B.S.D. Castro (CMUP) and is part of a project of using symmetries to obtain global stability results.