Good Reduction of Periodic Points on Projective Varieties

We consider dynamical systems arising from iterating a morphism of a projective variety defined over the field of fractions of a discrete valuation ring. Our goal is to obtain information about the dynamical system over the field of fractions by studying the dynamical system over the residue field. In particular, we aim to bound the possible primitive periods for a periodic point. This is discussed in dimension one with many references in [12, Section 2.6]. Recall that given a set A and map f : A → A we can create a dynamical system by iterating the map f on the set A. We denote f as the n iterate of f . An element a ∈ A such that f(a) = a for some positive integer n is called a periodic point and the least such n is called the primitive period of a. We will use the following notation unless otherwise specified: • R is a discrete valuation ring complete with respect to a normalized valuation v. Let m be the maximal ideal of R and let π be the uniformizer of m. • K is the field of fractions of R. • k = R/m is the finite residue field of characteristic p. • denotes reduction mod π. • | |v is the associated non-archimedean absolute value. Note that these conditions imply that K is a local field. In Section 2 we establish a notion of good reduction for a projective variety X/K and a morphism φ : X → X defined over K so that we can study the dynamics of φ over K by studying the dynamics of the reduced map over k. In Section 3 we describe the primitive period of P ∈ X(K) in terms of the primitive period the reduced point over k with the following two theorems.