An eigenvalue condition and the equivalence of two-dimensional maps

The condition on (Formula presented.) -maps of (Formula presented.) into itself is the assumption that their Jacobian eigenvalues are all equal to one (unipotent maps). A unipotent (Formula presented.) -map (Formula presented.) is equivalent to the translation (Formula presented.) if the map is fixed-point-free. It provides a one parameter family of (Formula presented.) -maps (Formula presented.) such that (Formula presented.) is linearly conjugated to G, (Formula presented.) has a global attractor for (Formula presented.) and a global repeller for (Formula presented.).