[Paper Review] Limit and end functors of dynamical systems via exterior spaces

J.M. García-Calcines, L.Hernandez Paricio+1 more|arXiv (Cornell University)|Feb 8, 2012

Advanced Differential Equations and Dynamical Systems1 references3 citations

TL;DR

This paper introduces a functorial framework for dynamical systems using exterior spaces, constructing limit and end spaces via absorbing open subsets (r-exterior sets) to analyze long-term behavior. The key contribution is proving that for locally compact T₃ spaces, the r-limit space equals the set of periodic points, and the bar-limit space equals the closure of the ω-limit set, unifying topological and dynamical invariants via category theory.

ABSTRACT

In this paper we analyze some applications of the category of exterior spaces to the study of dynamical systems (flows). We study the notion of an absorbing open subset of a dynamical system; i.e., an open subset that contains the "future part" of all the trajectories. The family of all absorbing open subsets is a quasi-filter which gives the structure of an exterior space to the flow. The limit space and end space of an exterior space is used to construct the limit spaces and end spaces of a dynamical system. On the one hand, for a dynamical system two limits spaces $L^{ }(X)$ and $\bar L^{ }(X)$ are constructed and their relations with the subflows of periodic, Poisson stable points and $Ω^{ }$-limits of $X$ are analyzed. On the other hand, different end spaces are also associated to a dynamical system having the property that any positive semi-trajectory has an end point in these end spaces. This type of construction permits us to consider the subflow containing all trajectories finishing at an end point $a$. When $a$ runs over the set of all end points, we have an induced decomposition of a dynamical system as a disjoint union of stable (at infinity) subflows.

Motivation & Objective

• To develop a categorical framework for studying dynamical systems using exterior spaces.
• To characterize the long-term behavior of flows through limit and end spaces derived from r-exterior subsets.
• To relate topological invariants like periodic points and ω-limits to functorial constructions in exterior homotopy theory.
• To establish a decomposition of flows into stable subflows via end space points.
• To generalize classical dynamical systems concepts using category-theoretic tools from proper homotopy theory.

Proposed method

• Define an r-exterior flow using absorbing open subsets that contain all future trajectories beyond some time.
• Construct the limit space $ L^\mathbf{r}(X) $ as the inverse limit over r-exterior sets $ E $, and $ \bar{L}^\mathbf{r}(X) $ as the inverse limit of their closures.
• Use the category of exterior spaces to define functors for limit and end spaces, enabling a functorial treatment of dynamical systems.
• Apply the notion of exterior homotopy theory to analyze the structure of phase spaces and their asymptotic behavior.
• Characterize points in $ \bar{L}^\mathbf{r}(X) \setminus L^\mathbf{r}(X) $ via frontier points of r-exterior sets.
• Utilize local compactness and regularity to relate topological properties to limit sets and closure of ω-limits.

Experimental results

Research questions

• RQ1 How can the category of exterior spaces be used to functorially describe the limit and end behavior of dynamical systems?
• RQ2 What is the relationship between the r-limit space $ L^\mathbf{r}(X) $ and the subflow of periodic points $ P(X) $?
• RQ3 How do the bar-limit space $ \bar{L}^\mathbf{r}(X) $ and the closure of the ω-limit set $ \overline{\Omega^\mathbf{r}(X)} $ relate in a locally compact T₃ space?
• RQ4 Can the end space of an exterior space be used to decompose a flow into stable subflows at infinity?
• RQ5 Under what topological conditions does $ \bar{L}^\mathbf{r}(X) = \overline{\Omega^\mathbf{r}(X)} $ hold?

Key findings

• For a locally compact $ T_3 $ space, the r-limit space $ L^\mathbf{r}(X) $ equals the subflow of periodic points $ P(X) $.
• The bar-limit space $ \bar{L}^\mathbf{r}(X) $ equals the closure of the $ \Omega^\mathbf{r} $-limit set $ \overline{\Omega^\mathbf{r}(X)} $ under the same conditions.
• The inclusion chain $ P(X) \subset P^\mathbf{r}(X) \subset \Omega^\mathbf{r}(X) \subset \overline{\Omega^\mathbf{r}(X)} = \bar{L}^\mathbf{r}(X) $ holds in locally compact $ T_3 $ spaces.
• A point $ x \in \bar{L}^\mathbf{r}(X) \setminus L^\mathbf{r}(X) $ exists if and only if there exists an r-exterior set $ E $ and time $ t $ such that $ t \cdot x \in \mathrm{Fr}(E) $.
• If $ x \notin \overline{\Omega^\mathbf{r}(X)} $, then there exists a neighborhood $ V_x $ such that $ X \setminus \overline{V}_x $ is r-exterior, implying $ x \notin \bar{L}^\mathbf{r}(X) $.
• In locally compact regular spaces, $ \bar{L}^\mathbf{r}(X) \subset \overline{\Omega^\mathbf{r}(X)} $, and equality holds under $ T_3 $ separation.