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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 is continuous and obeys $Φ_{s} (γ (t)) = γ (t + s)$ whenever defined (The fundamental theorem on flows).

Proof

technique · direct

1.1

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 $[T, \infty)$ .

given

2.1

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.

F1step 1.1

3.1

For fixed $s$ and any $T$ , [F2] sends $γ ([T, \infty))$ into $γ ([T + s, \infty))$ after increasing $T$ if necessary. Continuity therefore sends $ω (γ)$ into itself; applying the same argument to $- s$ gives equality.

F2step 2.1

4.1

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

step 1.1step 2.1step 3.1∎

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)