The Artin-Mazur Zeta Function of a Dynamically Affine Rational Map in Positive Characteristic

Abstract

A dynamically affine map is a finite quotient of an affine morphism of an algebraic group. We determine the rationality or transcendence of the Artin-Mazur zeta function of a dynamically affine selfmap of P 1 (k) for k an algebraically closed field of positive characteristic. The Artin-Mazur Zeta FunctionLet X be a set and let f : X → X define a dynamical system. Let Per n (f ) = {x ∈ X : f n (x) = x}, where f n denotes the composition of f with itself n times. The Artin-Mazur zeta function of this dynamical system is the formal power series given byAssume that # Per n (f ) < ∞ for all n, as otherwise ζ is not defined. The power series ζ(f, X; t) has rational coefficients, and it is not hard to show that ζ(f, X; t) ∈ Z [[t]] by means of the product formulaThis zeta function was introduced by Artin and Mazur in [2], where it is studied for X a manifold and f a diffeomorphism. In this setting, only the isolated periodic points are counted. This will not be an important distinction for our purposes.This paper continues the study of the following question, introduced in [4] for polynomials, but just as easily phrased for rational functions.The purpose of this paper is to answer this question for rational maps that are dynamically affine. These are maps that, loosely speaking, come from endomorphisms of algebraic groups; a precise definition will be given in section 2. There are five families of dynamically affine maps in one dimension:

Dynamically affine maps in positive characteristic

“…x → x 2 + 1 in characteristic p 5 (see [8,Question 2]). It would be interesting to investigate the nature of the (tame) zeta function for such examples.…”

Automorphism loci for degree 3 and degree 4 endomorphisms of the projective line

“…A 3 (C 4 ) and A 3 (D 4 ). From Silverman's classification of maps with cyclic automorphism groups [41,Proposition 7.3], we know that if a degree 3 map f has a C 4 symmetry, then f must be of the form 1 az 3 for some a ∈ Q. This family of maps is actually a family of twists corresponding to a single conjugacy class in…”

Bost-Connes systems and F1-structures in Grothendieck rings, spectra, and Nori motives

“…Remark 6.4. The properties of dynamical Artin-Mazur zeta functions change significantly when, instead of considering varieties over C one considers varieties in positive characteristic, [16], [19]. The prototype model of this phenomenon is illustrated by considering the Bost-Connes endomorphisms σ n : λ → λ n of G m ( Fp ).…”

“…The prototype model of this phenomenon is illustrated by considering the Bost-Connes endomorphisms σ n : λ → λ n of G m ( Fp ). In this case, the dynamical zeta function of σ n is rational or transcendental depending on whether p divides n (Theorem 1.2 and 1.3 and §3 of [16] and Theorem 1 of [17]). Similar phenomena in the more general case of endomorphisms of Abelian varieties in positive characteristic have been investigated in [19].…”