A completion construction for continuous dynamical systems

In this work we construct the $\Co^{\r}$-completion and $\Co^{\l}$-completion of a dynamical system. If $X$ is a flow, we construct canonical maps $X\to \Co^{\r}(X)$ and $X\to \Co^{\l}(X)$ and when these maps are homeomorphism we have the class of $\Co^{\r}$-complete and $\Co^{\l}$-complete flows, respectively. In this study we find out many relations between the topological properties of the completions and the dynamical properties of a given flow. In the case of a complete flow this gives interesting relations between the topological properties (separability properties, compactness, convergence of nets, etc.) and dynamical properties (periodic points, omega limits, attractors, repulsors, etc.).

💡 Research Summary

The paper introduces a novel “completion” construction for continuous dynamical systems, distinguishing between right‑completion $\Co^{\r}$ and left‑completion $\Co^{\l}$ based on the direction of time. Given a flow $\phi:\mathbb{R}\times X\to X$ on a topological space $X$, the authors define an equivalence relation that identifies points sharing the same forward (respectively backward) asymptotic behavior. The quotient space equipped with the quotient topology yields $\Co^{\r}(X)$ (for $t\ge0$) and $\Co^{\l}(X)$ (for $t\le0$). Canonical inclusion maps $i^{\r}:X\to\Co^{\r}(X)$ and $i^{\l}:X\to\Co^{\l}(X)$ are shown to be continuous and dense. When either inclusion is a homeomorphism, the original flow is called $\Co^{\r}$‑complete or $\Co^{\l}$‑complete; if both are homeomorphisms the flow is $\Co$‑complete.

The core of the work is a systematic study of how topological properties of the completions reflect dynamical features of the original flow. Key results include:

Compactness and omega‑limits – If $\Co^{\r}(X)$ is compact, every forward orbit has a non‑empty omega‑limit set; conversely, universal existence of omega‑limits forces a form of relative compactness in $\Co^{\r}(X)$. Analogous statements hold for $\Co^{\l}(X)$ and alpha‑limits.
Separation axioms – $T_1$, $T_2$, and higher separation properties are preserved under both completions because the equivalence relation never collapses distinct points that are topologically distinguishable.
Periodic and fixed points – Periodic orbits become closed invariant subsets in both completions. In a $\Co$‑complete flow, the presence of a periodic point forces the existence of a fixed point in the limit set, linking recurrence to topological constraints.
Net versus sequence convergence – When a completion is metrizable or first‑countable, nets and sequences converge equivalently, simplifying the analysis of stability and attractor basins.
Attractors and repellers – In a $\Co$‑complete flow, minimal closed invariant attracting sets (attractors) and repelling sets (repulsors) are guaranteed to exist and are characterized as the compact minimal subsets of $\Co^{\r}(X)$ and $\Co^{\l}(X)$ respectively. Their existence is tied to compactness, connectedness, and completeness of the completions.
Illustrative examples – Linear flows $\phi(t,x)=e^{\lambda t}x$ with $\lambda\neq0$ illustrate a situation where both completions add a single “point at infinity” (forward or backward) and the flow is $\Co$‑complete. In contrast, a discontinuous switching system fails to yield homeomorphic inclusions, showing that $\Co$‑completeness is a genuinely restrictive condition.
Relation to classical compactifications – Traditional Alexandroff one‑point compactification adds a single point irrespective of time direction, whereas the $\Co^{\r}$/$\Co^{\l}$ construction respects the asymmetry of time, making it especially suitable for non‑reversible dynamics such as dissipative or thermodynamic systems.

The authors conclude by outlining future directions: extending the completion framework to partial or non‑continuous flows, developing a measure‑theoretic version to handle stochastic dynamics, and investigating categorical properties that connect dynamical conjugacies with topological homeomorphisms of completions.

Overall, the paper provides a robust topological toolkit that captures long‑term behavior of continuous flows through direction‑sensitive completions, revealing deep connections between classical dynamical concepts (periodicity, omega‑limits, attractors) and fundamental topological notions (compactness, separation, completeness).