Monotone local flows with dense periodic orbits

TL;DR

This paper proves that in a partially ordered Euclidean space, a monotone local flow with dense periodic points must be globally periodic, revealing a strong link between local and global periodicity under these conditions.

Contribution

It establishes that monotone local flows with dense periodic points are necessarily globally periodic, a novel result connecting local behavior to global dynamics in ordered spaces.

Findings

Monotone local flows with dense periodic points are globally periodic.

The result applies to flows in Euclidean spaces with a convex cone order.

Provides a new criterion for global periodicity based on local properties.

Abstract

Let X be a connected open set in n-dimensional Euclidean space, partially ordered by a closed convex cone K with nonempty interior: y > x if and only if y-x is nonzero and in K. Theorem: If F is a monotone local flow in X whose periodic points are dense in X, then F is globally periodic.