  We consider sufficient conditions to determine the global dynamics for
equivariant maps of the plane with a unique fixed point which is also
hyperbolic. When the map is equivariant under the action of a compact Lie
group, it is possible to describe the local dynamics. In particular, if the
group contains a reflection, there is a line invariant by the map. This allows
us to use results based on the theory of free homeomorphisms to describe the
global dynamical behaviour. We briefly discuss the case when reflections are
absent, for which global dynamics may not follow from local dynamics near the
unique fixed point.
