An eigenvalue condition and the equivalence of two-dimensional maps

The condition on C1 -maps of R2 into itself is the assumption that their Jacobian eigenvalues are all equal to one (unipotent maps). A unipotent C1 -map G:R2→R2 is equivalent to the translation τ(x,y)=(x+1,y) if the map is fixed-point-free. It provides a one parameter family of C1 -maps Gμ:R2→R2 such that G0 is linearly conjugated to G, Gμ has a global attractor for ν>0 and a global repeller for ν<0 .