On topological properties of closed attractors

TL;DR

This work addresses the problem of when a closed (not necessarily compact) attractor $A$ in a locally compact metric space $X$ is homotopy equivalent to its domain of attraction $B_u(A)$ under uniform asymptotic stability. It develops a topological framework based on neighbourhood filters and $\mathscr{F}_d$-cofibrations to generalize known results from compact attractors to noncompact ones, proving that $A$ is a $\mathscr{F}_d$-weak deformation retract of $B_u(A)$ and giving conditions under which a strong deformation retract holds, linked to strong $\mathscr{F}_d$-neighbourhood deformation retracts or cofibrations. The paper further relates these findings to compact-attractor theory, discusses sets with positive reach as a practical sufficient condition for cofibration, and analyzes implications for linearization and feedback stabilization, including topological obstructions via extended cuts. The results provide a principled, topology-based perspective on designing feedback that achieves global stabilization of closed sets in noncompact settings, and they highlight cohomological constraints that arise in such stabilization tasks. Overall, the work connects homotopy-theoretic concepts with dynamical systems stability to illuminate fundamental limitations and possibilities for stabilization in noncompact spaces.

Abstract

The notion of an attractor has various definitions in the theory of dynamical systems. Under compactness assumptions, several of those definitions coincide and the theory is rather complete. However, without compactness, the picture becomes blurry. To improve our understanding, we characterize in this work when a closed, not necessarily compact, asymptotically stable attractor on a locally compact metric space is homotopy equivalent to its domain of attraction. This enables a further structural study of the corresponding feedback stabilization problem.

Figures (3)

• Figure 1.1: Examples of a closed, but non-compact, attractors, with in $(i)$$A$ being of the form $\mathbb{R}^2\setminus \{x:\|x\|_2<1\}$ such that $\partial A =\mathbb{S}^1$, whereas in $(ii)$ the underlying space $X$ is of the form $\mathbb{R}^2\setminus \{0\}$ such that $A$ as drawn is closed. In $(iii)$$X$ is $\mathbb{R}^2$ with $\{ (1,0),(0,1),(-1,0),(0,-1)\}$ removed, this is slightly more involved version of the example in Section \ref{['sec:correct:Wilson']}. At last, in $(iv)$ one sees how closed attractors might emerge by grouping several invariant sets.
• Figure 2.1: Example \ref{['ex:open:nbhd:closed:1']}: for closed, but non-compact, subsets $A$ of a metric space $X$ it is not true that any open neighbourhood $U$ of $A$ contains a metric neighbourhood $N_{\varepsilon}$ of $A$.
• Figure 3.1: Example \ref{['ex:counterS1']} and Example \ref{['ex:S1closed']}. The set $A$ is a uniformly asymptotically stable attractor, yet, $A$ is not a strong deformation retract of $B_u(A)$. A (partial) reason being that $\mathscr{F}_{\tau}\not \leq \mathscr{F}_d$, e.g., $\not\exists V\in \mathscr{F}_d:V\subseteq U$.

Theorems & Definitions (40)

• Definition 2.1: $\mathscr{F}$-weak deformation retract
• Definition 2.2: $\mathscr{F}$-neighbourhood (deformation) retract
• Lemma 2.3: Coarser and finer filter retracts
• proof
• Lemma 2.4: A strong deformation retract factored through filters
• proof
• Lemma 2.5: On $\varepsilon$-neighbourhoods
• proof
• Example 2.6: Open neighbourhoods of closed subsets 1
• Definition 2.7: $\mathscr{F}$-cofibration