LemmaStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedaudited 2026-09-06

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Stable and unstable manifolds are flow invariant

Statement

For every $s \in R$ and critical point $p$ of a complete descending flow $Φ$ ,

$Φ_{s} (W^{s} (p)) = W^{s} (p), Φ_{s} (W^{u} (p)) = W^{u} (p) .$

Facts & Assumptions

Given: A complete descending flow $Φ$ , a critical point $p$ , and $s \in R$ .

[F1]

Stable and unstable sets are the forward and backward convergence sets (Stable and unstable sets of a critical point).

[F2]

The flow satisfies $Φ_{t} (Φ_{s} (x)) = Φ_{t + s} (x)$ and has inverse $Φ_{- s}$ (The fundamental theorem on flows).

Proof

technique · direct

1.1F1F2given

If $x \in W^{s} (p)$ , then [F2] gives $Φ_{t} (Φ_{s} (x)) = Φ_{t + s} (x) \to p$ as $t \to \infty$ , so $Φ_{s} (x) \in W^{s} (p)$ by [F1]. The same calculation with $t \to - \infty$ proves inclusion for $W^{u} (p)$ .

2.1F2step 1.1

Applying step 1.1 with $- s$ and using the inverse in [F2] proves the reverse inclusions.

3.1step 1.1step 2.1∎

Therefore both stable and unstable sets are invariant under every fixed flow time.

Depends on

Used by

Dependency tree · two levels

6 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

Ralph L. Cohen, Bundles, Manifolds, and Homotopy, §13.2 (standard reference, not scraped)