In this paper we expand the theory of connected components in non-archimedean discrete dynamical systems. We define two types of components and discuss their uses and applications in the study of dynamics of a rational function $\phi\in K(z)$ defined over a non-archimedean field K. We prove that some fundamental conjectures, including the No Wandering Domains conjecture, are equivalent, regardless of which definition of 'component' is used. We derive several results on the geometry of our components and the existence of periodic points within them. We also give a number of examples of p-adic maps with interesting or pathological dynamics.

2000 Mathematical Subject Classification: primary 37B99; secondary 11S99, 30D05.