Rational Periodic Points for Degree Two Polynomial Morphisms on Projective Space

TL;DR

This paper constructs infinite families of degree 2 morphisms on projective space with rational periodic points of large primitive period, demonstrating that such periods can grow faster than any polynomial in the dimension.

Contribution

The authors explicitly construct morphisms with arbitrarily large rational periodic points, showing that the primitive period can grow faster than any polynomial in the dimension for large N.

Findings

Existence of morphisms with large rational periodic points

Primitive period can grow faster than polynomial in dimension

Construction of explicit infinite families of morphisms

Abstract

This article addresses the existence of $\Q$-rational periodic points for morphisms of projective space. In particular, we construct an infinitely family of morphisms on $\P^N$ where each component is a degree 2 homogeneous form in $N+1$ variables which has a $\Q$-periodic point of primitive period $\frac{(N+1)(N+2)}{2} + \lfloor \frac{N-1}{2}\rfloor$. This result is then used to show that for $N$ large enough there exists morphisms of $\P^N$ with $\Q$-rational periodic points with primitive period larger that $c(k)N^k$ for any $k$ and some constant $c(k)$.