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Precompact trajectory tails have nonempty compact connected flow-invariant limit sets

Statement

Let $γ : R \to M$ be a full negative-gradient trajectory and let $Φ$ be its flow. If $\overline{γ ([0, \infty))}$ is compact, then

$ω (γ) : = ⋂_{T \geq 0} \overline{γ ([T, \infty))}$

is nonempty, compact, connected, and invariant under every $Φ_{s}$ . The analogous conclusion holds for $α (γ) : = ⋂_{T \geq 0} \overline{γ ((- \infty, - T])}$ when its negative tail has compact closure.

Facts & Assumptions

Given: A full trajectory $γ$ and a compact closure $K$ of its positive tail.

[F1]

A compact space has the finite-subcover property (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).

[F2]

The maximal flow has open domain in $R \times M$ , is continuous, and obeys $Φ_{s} (γ (t)) = γ (t + s)$ whenever defined (The fundamental theorem on flows).

Proof

technique · direct

1.1given

Each $K_{T} : = \overline{γ ([T, \infty))}$ is a nonempty closed subset of $K$ , the family is decreasing, and each $K_{T}$ is connected because it is the closure of the connected image of $[T, \infty)$ .

1.2F1F2given

The flow domain is open and contains ${0} \times K$ by [F2]. For each $q \in K$ there are $δ_{q} > 0$ and a neighbourhood $V_{q}$ with $(- δ_{q}, δ_{q}) \times V_{q}$ in that domain. A finite subcover of $K$ gives one $δ > 0$ such that $Φ_{s} (p)$ is defined for every $p \in K$ and $∣ s ∣ < δ$ .

2.1F1step 1.1

If $⋂_{T \geq 0} K_{T}$ were empty, the open sets $K ∖ K_{T}$ would cover $K$ ; [F1] would give finitely many of them that cover. Since the $K_{T}$ decrease, one already covers, contradicting $K_{T} \neq \emptyset$ . Thus $ω (γ)$ is nonempty; it is closed in $K$ , hence compact, and the nested connected-set argument makes it connected.

3.1F2step 2.1step 1.2

Let $p \in ω (γ)$ and $∣ s ∣ < δ$ . Fix $T \geq 0$ and a neighbourhood $W$ of $Φ_{s} (p)$ . Continuity gives a neighbourhood $V$ of $p$ with $Φ_{s} (V) \subseteq W$ . Since $p \in K_{T + ∣ s ∣}$ , some $t \geq T + ∣ s ∣$ has $γ (t) \in V$ ; then $γ (t + s) = Φ_{s} (γ (t)) \in W$ and $t + s \geq T$ . Thus $Φ_{s} (p) \in K_{T}$ for every $T$ , so $Φ_{s} (ω (γ)) \subseteq ω (γ)$ . Divide any real $s$ into finitely many increments of size less than $δ$ and apply this inclusion repeatedly; all intermediate points remain in $ω (γ) \subseteq K$ , where the next increment is defined. The flow law then defines $Φ_{s}$ on all of $ω (γ)$ and maps it into itself. Applying the argument to $- s$ gives equality.

4.1step 1.1step 2.1step 1.2step 3.1∎

Replacing $t$ by $- t$ and using the reverse-time flow gives the asserted nonempty compact connected invariant set $α (γ)$ for a precompact negative tail.

Depends on

Used by

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Liviu I. Nicolaescu, An Invitation to Morse Theory, Lemma 2.4.1 (standard reference, not scraped)