Dynamics of maps with nilpotent Jacobians

Abstract

The Discrete Markus-Yamabe Conjecture (also known as the LaSalle Conjecture) imposed conditions on the Jacobian eigenvalues of a map in the hope of ensuring global attractivity of the fixed point. This paper pushes such assumptions to their extreme; the Jacobian is assumed to be nilpotent at all points. The dynamics of such maps is studied and diverse behaviour is observed, from the quick collapse of points to a globally attractive fixed point, to maps with self-intersecting invariant curves.