A criterion for potentially good reduction in nonarchimedean dynamics

Abstract

Let K be a non-archimedean eld, and let 2 K(z) be a polynomial or rational function of degree at least 2. We present a necessary and sucient condition, involving only the xed points of and their preimages, that determines whether or not the dynamical system : P 1 ! P 1 has potentially good reduction. Fix the following notation throughout this paper. K: a eld K: an algebraic closure of K j j v: a non-archimedean absolute value on K OK: the ring of integersfx2 K :jxjv 1g of K MK: the maximal ideafx2 K :jxjv < 1g ofOK k: the residue eld