Projective Cyclic Groups in Higher Dimensions

Abstract: In this article we provide a classification of the projective transformations in $PSL(n+1,\Bbb{C})$ considered as automorphisms of the complex projective space $\Bbb{P}^n$. Our classification is an interplay between algebra and dynamics, which just as in the case of isometries of CAT(0)-spaces, can be given by means of tree three types, namely: elliptic, parabolic and loxodromic. We carefully describe the dynamic in each case, more precisely we determine the corresponding Kulkarni's limit set, the equicontinuity region, the discontinuity region and in some cases we provide families of maximal regions where the respective cyclic group acts properly discontinuously. Also we provide, in each case, some equivalents ways to classify the projective transformations.