Monotone dynamical systems with dense periodic points

TL;DR

This paper proves that in certain monotone dynamical systems, the density of periodic points implies the system itself is periodic, confirming a conjecture by M. Hirsch.

Contribution

It establishes that a discrete monotone dynamical system with dense periodic points must be globally periodic, confirming a longstanding conjecture.

Findings

Periodic points are dense in the system.

The system is proven to be globally periodic.

Supports the conjecture by M. Hirsch.

Abstract

In this paper we prove a recent conjecture by M. Hirsch, which says that if $(f,\Omega)$ 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 $\Omega$, then $f$ is periodic.