Priestley duality and representations of recurrent dynamics

TL;DR

This paper explores the relationship between global dynamics of systems and their order-theoretic representations via Priestley duality, offering a new framework for analyzing complex dynamical behaviors.

Contribution

It introduces an order-theoretic framework connecting global dynamics with Priestley spaces, extending classical concepts to arbitrary topological spaces.

Findings

Priestley duality provides a new perspective on dynamical systems.

The framework generalizes the chain recurrent set as a Priestley space.

Offers a Hausdorff compactification of the recurrent set.

Abstract

For an arbitrary dynamical system there is a strong relationship between global dynamics and the order structure of an appropriately constructed Priestley space. This connection provides an order-theoretic framework for studying global dynamics. In the classical setting, the chain recurrent set, introduced by C. Conley, is an example of an ordered Stone space or Priestley space. Priestley duality can be applied in the setting of dynamics on arbitrary topological spaces and yields a notion of Hausdorff compactification of the (chain) recurrent set.