Wandering Flows on the Plane

TL;DR

This work analyzes wandering (non-recurrent) planar flows, restricting to regular flows, and shows that on the plane such flows have no generalized recurrence and are classified by exactly two invariants: the topology of the orbit space and the prolongational (Auslander) stream ${\cal A}_F$. It establishes that all prolongational limit sets stabilize at order 2, i.e., $\Lambda^2_F(p)=\bigcup_{k\in\mathbb{N}}\Lambda^{1,k}_F(p)$, and that the generalized recurrent set $A_F$ is empty, guaranteeing the existence of a strict $C^0$ Lyapunov function. The authors leverage Kaplan’s chordal framework to prove a complete topological classification: two regular planar flows are equivalent iff their Auslander streams are isomorphic under a plane homeomorphism, meaning the two invariants fully determine the flow up to topological equivalence. These results illustrate that the stream encapsulates fundamental dynamical information even in the absence of recurrence and connect classical foliations theory with modern prolongational concepts.

Abstract

We study planar flows without non-wandering points and prove several properties of these flows in relation with their prolongational relation. The main results of this article are that a planar (regular) wandering flow has no generalized recurrence and has only two topological invariants: the space of its orbits and its prolongational relation (or, equivalently, its smallest stream). As a byproduct, our results show that, even in absence of any type of recurrence, the stream of a flow contains fundamental information on its behavior.

Figures (6)

• Figure 1: An example of a family of continuous mutually disjoint curves filling the plane. No point on $s_\infty$ has a neighborhood where the family is topologically equivalent to a family of parallel straight lines.
• Figure 2: An example of a flow with a non-closed set of separatrices. Separatrices are painted in blue, cyan and red. Orbits that are not separatrices are painted in green.
• Figure 3: (left) Sketch of the flow $F$ of the vector field $\xi(x,y)=(2y,1-y^2)$. The two separatrices of $F$ are painted in blue. (right) Sketch of a flow $F$ with four separatrices, painted in blue.
• Figure 4: (left) Sketches of the flows of the vector fields $\xi(x,y)=(\sin y,\cos y)$ (left) and $\xi(x,y)=(\sin y,\cos^2 y)$ (right). Their separatrices are painted in blue.
• Figure 5: Pictures showing three orbits $\alpha,\beta,\gamma$ with $|\alpha,\beta,\gamma|^+$ under different configurations. The yellow arcs meet each orbit in at most a point. Each other colored line is an orbit of the flow. (a) Each orbit is reachable from each of the other two. The picture suggests that such configuration cannot happen for a regular planar flow. (b,c) $\alpha$ is reachable from $\beta$ and $\gamma$ while $\beta$ is not reachable from $\gamma$. Under this configuration, there must exist inseparable separatrices $\sigma,\sigma'$ (painted in gold) and there are two inequivalent cases shown in panels (b) and (c). (d) $\gamma$ is not reachable neither from $\alpha$ nor from $\beta$. Under this configuration there are several cases. All reduce to either (b) or (c) except for the one shown in this panel.

Theorems & Definitions (122)

• Theorem A: Kaplan, 1940 Kap40
• Theorem B: Kaplan, 1948 Kap48
• Theorem C: Whitney, 1933 Whi33, Theorem 27A; Kaplan, 1940 Kap40, Corollary to Theorem 42
• Theorem D: Whitney, Kaplan
• Definition 2.1
• Example 2.2
• Definition 2.3
• Lemma 2.4: De Leo and Yorke DLY24
• Proposition 2.5: De Leo and Yorke DLY24
• Definition 2.6