# Dynamically affine maps in positive characteristic

## TL;DR

This paper investigates the fixed points and zeta functions of dynamically affine maps over algebraically closed fields of positive characteristic, revealing a dichotomy in their rationality and algebraicity properties.

## Contribution

It introduces hypotheses that determine when the Artin-Mazur zeta function is rational or non-holonomic and verifies these for specific classes of maps, extending prior results.

## Key findings

- Zeta function is either rational or non-holonomic depending on the map.
- Hypotheses are verified for maps on the projective line and Kummer varieties.
- Algebraicity of the tame zeta function is studied in this context.

## Abstract

We study fixed points of iterates of dynamically affine maps (a generalisation of Latt\`es maps) over algebraically closed fields of positive characteristic $p$. We present and study certain hypotheses that imply a dichotomy for the Artin-Mazur zeta function of the dynamical system: it is either rational or non-holonomic, depending on specific characteristics of the map. We also study the algebraicity of the so-called tame zeta function, the generating function for periodic points of order coprime to $p$. We then verify these hypotheses for dynamically affine maps on the projective line, generalising previous work of Bridy, and, in arbitrary dimension, for maps on Kummer varieties arising from multiplication by integers on abelian varieties.