Projective cyclic groups in higher dimensions

In this article we provide a classification of the projective transformations in PSL(n+1,C) considered as automorphisms of the complex projective space PCn. Our classification is an interplay between algebra and dynamics. Just as in the case of isometries of CAT(0)-spaces, this is given by means of three types of transformations, namely: elliptic, parabolic and loxodromic. We describe the dynamics in each case, more precisely we determine the corresponding Kulkarni's limit set, the equicontinuity region, minimal sets, the discontinuity region and maximal regions where the corresponding cyclic group acts properly discontinuously. We also provide, in each case, some equivalent ways to classify the projective transformations.