Monotone Dynamical Systems with Polyhedral Order Cones and Dense Periodic Points

TL;DR

This paper proves that in a connected, dense subset of R^n ordered by a polyhedral cone, any monotone homeomorphism with dense periodic points must itself be periodic, extending understanding of dynamical systems with order structure.

Contribution

It establishes that monotone homeomorphisms with dense periodic points are necessarily periodic in the setting of polyhedral cone-ordered subsets of R^n.

Findings

Monotone homeomorphisms with dense periodic points are periodic.

The result applies to subsets of R^n with connected, dense interior.

The ordering is defined by a polyhedral cone with nonempty interior.

Abstract

Let X be a subset of R^n whose interior is connected and dense in X, ordered by a polyhedral cone in R^n with nonempty interior. Let T be a monotone homeomorphism of X whose periodic points are dense. Then T is periodic.