A note on periodic points of order preserving subhomogeneous maps

Let $\mathbb{R}_+^n$ be the standard closed positive cone in $\mathbb{R}^n$ and let $\Gamma(\mathbb{R}_+^n)$ be the set of integers $p\geq 1$ for which there exists a continuous, order preserving, subhomogeneous map $f\colon \mathbb{R}_+^n\to \mathbb{R}_+^n$, which has a periodic point with period $p$. It has been shown by Akian, Gaubert, Lemmens, and Nussbaum that $\Gamma(\mathbb{R}_+^n)$ is contained in the set $B(n)$ consisting of those $p\geq 1$ for which there exist integers $q_1$ and $q_2$ such that $p=q_1q_2$, $1\leq q_1\leq {n\choose m}$, and $1\leq q_2\leq {m\choose \lfloor m/2\rfloor}$ for some $1\leq m\leq n$. This note shows that $\Gamma(\mathbb{R}_+^n)=B(n)$ for all $n\geq 1$.