[Paper Review] Chain Recurrence For General Spaces

Ethan Akin, Jim Wiseman|arXiv (Cornell University)|Jul 30, 2017

Advanced Topology and Set Theory5 references3 citations

TL;DR

This paper extends the theory of chain recurrence to general topological spaces and relations, generalizing Conley and Aubry-Mather chain relations beyond metric spaces. It establishes foundational results using barrier functions, Lyapunov functions, and proper maps, proving that the chain relations are closed and invariant under proper dynamics, with key results on compactness and continuity in uniform and k-spaces.

ABSTRACT

The chain relation, due to Conley, and the strong chain relation, due to Easton, are well studied for continuous maps on compact metric spaces. Following Fathi and Pageault, we use barrier functions to generalize the theory to general relations on uniform spaces. In developing the theory, we indicate why the chain ideas are naturally uniform spaces concepts. We illustrate that the extension to relations is easy and is useful even for the study of the continuous map case.

Motivation & Objective

• To generalize the concepts of Conley and Aubry-Mather chain relations from metric spaces to general topological spaces and relations.
• To establish the theoretical foundations for chain recurrence in non-metric, non-compact settings using relations and uniform structures.
• To investigate the topological and dynamical properties of chain relations under proper maps and compactifications.
• To clarify the role of barrier and Lyapunov functions in characterizing chain recurrence in general spaces.
• To prove that chain relations are closed and invariant under proper dynamics in k-spaces and uniform spaces.

Proposed method

• Extends chain recurrence to relations on topological spaces by defining ε-chains and strong ε-chains using pseudo-metrics.
• Introduces the Conley chain relation $\mathcal{C}f$ as the set of pairs $(x,y)$ with ε-chains from $x$ to $y$ for all $\epsilon > 0$, and the Aubry-Mather relation $\mathcal{A}_d f$ using strong ε-chains.
• Uses barrier functions $M_d^f(x,y)$ and Lyapunov functions $L_d^f(x,y)$ to characterize the chain relations as the infimum of ε for which chains exist.
• Applies the theory of proper maps and compactifications to ensure closedness and invariance of chain relations in Tychonoff and k-spaces.
• Employs nets and directed sets to handle convergence and cluster points in general topological spaces.
• Utilizes uniform space structures to generalize metric-based results, particularly for closedness and continuity of chain relations.

Experimental results

Research questions

• RQ1 How can the Conley and Aubry-Mather chain relations be generalized from metric spaces to arbitrary topological spaces and relations?
• RQ2 What conditions ensure that the chain relations $\mathcal{C}f$ and $\mathcal{A}_df$ are closed in general topological spaces?
• RQ3 Under what conditions are the chain relations invariant under proper dynamics?
• RQ4 How do barrier and Lyapunov functions characterize chain recurrence in non-metric settings?
• RQ5 What is the role of compactness and proper maps in ensuring the topological closure of chain relations?

Key findings

• The Conley chain relation $\mathcal{C}f$ is independent of the choice of metric, while the Aubry-Mather relation $\mathcal{A}_d f$ depends on the pseudo-metric $d$.
• $\mathcal{C}f$ is closed in $X \times X$ when $f$ is a proper map on a Tychonoff space, and $\mathcal{A}_d f$ is closed under the same conditions.
• If $f$ is a proper map on a Tychonoff space, then $\mathcal{C}f$ is $f$-invariant, meaning $f(\mathcal{C}f) \subset \mathcal{C}f$.
• For a proper map $f$ on a Tychonoff space, the chain relations $\mathcal{C}f$ and $\mathcal{A}_d f$ are closed in $X \times X$.
• In a k-space, if $f$ is proper and $B \subset Y$ is compact, then $f^{-1}(B)$ is compact, ensuring the preimage of compact sets is compact.
• The composition of proper maps is proper, and proper maps preserve compactness in preimages, which is essential for proving closedness of chain relations.