On linearly ordered sets of chain components

TL;DR

This paper explores the order-theoretic structure of chain components in dynamical systems, revealing constraints and possibilities in their partial orderings under various regularity conditions.

Contribution

It characterizes the possible order types of chain component posets in continuous and non-regular dynamical systems, linking order theory with dynamical behavior.

Findings

Chain components poset cannot be linearly and densely ordered if f is continuous.

Every countable well-order with a maximum can be realized as the chain components poset of an interval map.

There exists a dynamical system with a countable densely ordered chain components poset.

Abstract

In a dynamical system $(X,f)$, with $X$ a compact metric space, the chain components, the fundamental building blocks in the Conley decomposition of dynamics, have a natural partial order induced by the chain relation between points. Although chain components are crucial for understanding the long-term behavior of topological systems, they have not been widely studied from the point of view of poset theory. In this work, we pursue this line of research, considering both the case in which $f$ is a continuous map and the general case in which no regularity assumption is made. Our main results are that, if $f$ is continuous: - the chain components poset cannot be linearly and densely ordered; - every countable well-order with a maximum is the order type of the chain components poset of an interval map. If no regularity assumption is made: - there is a dynamical system on the interval whose chain components poset is countable and densely ordered; - the chain components poset has at least one minimal element. These results, bridging dynamical systems and order theory, highlight both the structural constraints and the possibilities for the chain components posets.