Iteration of Rational Functions in Positive Characteristic

Abstract

In arithmetic dynamics we study the iteration of a map f on a set S with a certain number-theoretical meaning. We use the dynamical zeta function, which encodes all information on the (finite) number of fixed points of all n-th iterates of f on S. We are particularly interested in the case where S consists of all points of an algebraic variety X over the algebraic closure of the ground field.

If X is the projective line over a field K of characteristic zero, and f is a rational map of degree at least 2, then the dynamical zeta function is a rational function over Q(T). However, if K has positive characteristic, then the dynamical zeta function of so-called dynamically affine maps, which are morphisms of a strongly group-theoretical nature, becomes transcendental over Q(T).

Under some assumptions, we prove new results for separable endomorphisms on an elliptic curve E over a field K of characteristic p>0, and for multiplication-by-m maps on abelian varieties, where p does not divide m.

Such transcendence results indicate that for characteristic p>0 the number of fixed points does not have an easy pattern. The tame dynamical zeta function is introduced as an alternative for the original dynamical zeta function; it only counts n-th iterates for p not dividing n. We prove a new theorem which tells us that for dynamically affine maps, the tame dynamical zeta function is algebraic.