Rational Periodic Points for Degree Two Polynomial Morphisms on Projective Space

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