  It is demonstrated that a C^1-unipotent map is globally equivalent to the
linear translation T(x,y)=(x+1,y), if the map is fixed point free Similarly, it
is proved not only that the fixed point set induced by a C^1-unipotent has no
isolated elements, but that a $C^1-$unipotent map has no periodic points. The
relation with the existence of global attractors in R^2, by using a global
bifurcation on unipotent maps, is also studied.
