Monotone dynamical systems with dense periodic points

In this paper we prove a recent conjecture by M. Hirsch, which says that if $(f,\mathrm{\Omega })$ is a discrete time monotone dynamical system, with $f\colon \Omega\to\Omega$ a homeomorphism on an open connected subset of a finite dimensional vector space, and the periodic points of $f$ are dense in $\mathrm{\Omega }$, then $f$ is periodic.