A geometric approach to divergent points of higher dimensional Collatz mappings

We define generalized Collatz mappings on free abelian groups of finite rank and study their iteration trajectories. Using geometric arguments we describe cones of points having a divergent trajectory and we deduce lower bounds for the density of the set of divergent points. As an application we give examples which show that the iteration of generalized Collatz mappings on rings of algebraic integers can behave quite differently from the conjectured behavior in $Z$.