LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Translate every finite forward orbit without assuming global invertibility

Statement

Use the translation $τ (y) = p + y$ and conjugate $f (y) = h (p + y) - p$ of the fixed-point translation lemma. For every integer $n \geq 0$ , the iterates through time $n$ are defined at $y$ for $f$ exactly when they are defined at $p + y$ for $h$ , and then $h^{n} (p + y) = p + f^{n} (y)$ . An iterate of a local map is defined only if all preceding points lie in its domain; $h^{0}$ and $f^{0}$ are identities. Consequently, for any $B \subseteq U - p$ , every forward iterate of $y$ lies in $B$ exactly when every forward iterate of $p + y$ lies in $p + B$ .

Facts & Assumptions

[F1]

The conjugacy is defined on $U - p$ and obeys $h \circ τ = τ \circ f$ Translate a local diffeomorphism and its spectral data to the origin.

[F2]

Mathematical induction: if a property holds at $0$ and its validity at $n$ implies its validity at $n + 1$ for every $n \geq 0$ , it holds for all nonnegative integers The principle of mathematical induction.

Proof

Given: The maps, domains and iterate convention in the Statement.

Goal: Establish equality of the orbit conditions, including their domains.

1.1givenF1F2

At time zero the equality is $p + y = p + y$ . If it holds through time $n$ , then $f^{n} (y) \in U - p$ exactly when $h^{n} (p + y) \in U$ . These are precisely the conditions for the next iterate to be defined, and conjugacy gives $h^{n + 1} (p + y) = p + f^{n + 1} (y)$ . Induction proves the claim for every finite $n$ .

2.1step 1.1∎

Applying the finite identity for each $n \geq 0$ gives $f^{n} (y) \in B$ exactly when $h^{n} (p + y) \in p + B$ . Requiring these equivalent conditions for every $n$ proves the assertion about complete forward orbits.

Depends on

Used by · 1

Transport local invariance and forward trapping to a fixed point

Dependency tree · 2 levels

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