Components and Periodic Points in Non-Archimedean Dynamics

In this paper we expand the theory of connected components innon-archimedean discrete dynamical systems. We define two typesof components and discuss their uses and applications in thestudy of dynamics of a rational function K(z) defined overa 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 deriveseveral results on the geometry of our components and the existenceof periodic points within them. We also give a number of examplesof p-adic maps with interesting or pathological dynamics. 2000Mathematical Subject Classification: primary 37B99; secondary11S99, 30D05.