Integral periodic orbits on affine spaces

TL;DR

This paper establishes a universal upper bound on the size of integral periodic orbits for endomorphisms in affine spaces, providing explicit formulas and demonstrating the bound in the affine plane.

Contribution

It introduces an elementary proof for a uniform upper bound on integral periodic orbits, linking primitive periods to local reductions, with a specific bound of 24 in the affine plane.

Findings

Bound of 24 for integral periodic orbits in the affine plane

Formula relating primitive period to local primitive period

Effective uniform upper bound dependent only on dimension

Abstract

In this paper, we give an elementary proof on the existence of an effective uniform upper bound on the size of integral periodic orbits of a single endomorphism in an affine space, dependent solely on its dimension. In fact, we derive a formula relating the primitive period to the local primitive period obtained through reduction modulo prime number. In particular, we prove that the size of any integral periodic orbit in the affine plane does not exceed 24.