A Glance into the Anatomy of Monotonic Maps

TL;DR

This paper explores conditions under which autohomeomorphisms on ordered topological spaces can be re-ordered to become monotonic, linking such re-orderings to topological conjugacy with monotonic maps.

Contribution

It introduces criteria for reordering spaces to make maps monotonic and connects this to conjugacy, expanding understanding of monotonic maps in ordered topological spaces.

Findings

Reordering spaces can make certain maps monotonic.

Existence of reordering is equivalent to conjugacy to a monotonic map.

Not all re-orderings are order-isomorphic to the original space.

Abstract

Given an autohomeomorphism on an ordered topological space or its subspace, we show that it is sometimes possible to introduce a new topology-compatible order on that space so that the same map is monotonic with respect to the new ordering. We note that the existence of such a re-ordering for a given map is equivalent to the map being conjugate (topologically equivalent) to a monotonic map on some homeomorphic ordered space. We observe that the latter cannot always be chosen to be order-isomorphic to the original space. Also, we identify other routes that may lead to similar affirmative statements for other classes of spaces and maps.