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On linearly ordered sets of chain components

P. Cintioli, A. Della Corte, M. Farotti

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.

On linearly ordered sets of chain components

Abstract

In a dynamical system , with 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 is a continuous map and the general case in which no regularity assumption is made. Our main results are that, if 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.

Paper Structure

This paper contains 6 sections, 15 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Linearly ordered structures in $\mathcal{C}$-posets of continuous systems
  3. Compact spaces
  4. Non-compact spaces
  5. $\mathcal{C}$-posets in discontinuous systems
  6. $\mathcal{C}^+$-posets under topological conjugacy

Key Result

Theorem 1

Let $X,Y$ be topological spaces and $f: X\rightarrow Y$ continuous function. Let $\sim$ be an equivalence relation on $X$ and $\pi : X\rightarrow X/\sim$ the canonical quotient map. There exists a continuous map such that $g \circ \pi =f$ if and only if $f$ is constant on equivalence classes.

Theorems & Definitions (44)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Definition 4
  • Remark 2
  • Definition 5
  • Definition 6
  • Definition 7
  • Theorem 1
  • ...and 34 more