Dynamics in Dimension Greater Than One

Up to this point our primary focus has been on arithmetic dynamics of rational maps on ℙ1. In this chapter we take a look at dynamics in higher dimensions. Even over ℂ, although there is now a significant body of knowledge, it seems fair to say that complex dynamics on ℙ N is still in its infancy. And arithmetic dynamics in higher dimensions is at present a patchwork of results from which a general theory is yet to emerge. Our goal in this chapter is to provide a glimpse into two aspects of this developing theory by highlighting two ways in which higher-dimensional dynamics differs significantly from the one-dimensional case.