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 a local diffeomorphism and its spectral data to the origin

Statement

Let $h : U \to R^{m}$ be a $C^{r}$ local diffeomorphism, $r \geq 1$ a finite integer, with $U$ open and $h (p) = p$ . Here local diffeomorphism means that around each point the map has a $C^{r}$ inverse onto an open image. Set $τ (y) = p + y$ , $V = U - p$ , and $f = τ^{- 1} \circ h \circ τ$ . Then $f : V \to R^{m}$ is a $C^{r}$ local diffeomorphism fixing zero, $D f (0) = D h (p)$ , and the generalized eigenspaces of these two matrices coincide. For a real matrix, spectral subspaces for nonreal eigenvalues mean the real forms of the conjugation-invariant sums in its complexification. Here $C^{r}$ uses continuous iterated Fréchet derivatives through order $r$ . The complexification has canonical conjugation $σ (z \otimes v) = \overline{z} \otimes v$ , and the real form of an invariant complex subspace $W$ means $W^{σ} = {w \in W : σ (w) = w}$ , identified with a subspace of $R^{m}$ via $v \mapsto 1 \otimes v$ .

Facts & Assumptions

[F1]

The total derivative obeys the Euclidean chain rule The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a).

[F2]

Finite $C^{r}$ regularity means that the iterated Fréchet derivatives exist and are continuous in their iterated operator norms C k map between Banach spaces. Fréchet differentiability means a bounded linear approximation with a remainder whose norm divided by the increment norm tends to zero Fréchet derivative between Banach spaces. These definitions apply to $R^{m}$ and its Banach spaces of iterated derivative values.

[F3]

Generalized eigenspaces are the kernels $\ker (T - λ I)^{k}$ Primary components ker q(T)ᵉ and generalised eigenspaces G_λ⁽ᵉ⁾(T)=ker(T-λ I)ᵉ.

[F4]

The complexification is $C \otimes_{R} R^{m}$ with real embedding $v \mapsto 1 \otimes v$ Complexification as ℂ⊗_ℝV with its canonical real-linear embedding, and a real operator $T$ extends as $T_{C} (z \otimes v) = z \otimes T (v)$ Complexification of a real-linear map.

[F5]

The canonical conjugation is $σ (z \otimes v) = \overline{z} \otimes v$ Conjugations and real structures on a complex vector space. The fixed real form of a conjugation-invariant subspace $W$ is $W^{σ}$ The fixed real form of a conjugation.

Proof

Given: The map, fixed point, finite integer regularity and conventions in the Statement.

Goal: Establish the translated local diffeomorphism and unchanged spectral data.

1.1givenF2

Translation $τ$ has inverse $x \mapsto x - p$ ; both preserve distances, so they carry each open ball to an open ball of the same radius and preserve openness. In particular $V$ is open. Their first derivatives are identity maps and their higher derivatives vanish, as follows from the zero remainders for affine maps and then differentiating a constant derivative.

2.1F2step 1.1

To check every required derivative, let $H : O \to R^{m}$ be any $C^{r}$ map on an open set, and set $g (y) = H (a + y) + b$ on $O - a$ . Its differentiability remainder equals $H (a + y + u) - H (a + y) - D H (a + y) u$ , so $D g (y) = D H (a + y)$ . Inductively, if $D^{j} g (y) = D^{j} H (a + y)$ for $1 \leq j < r$ , apply the same remainder identity to the Banach-valued map $D^{j} H$ to obtain $D^{j + 1} g (y) = D^{j + 1} H (a + y)$ . Continuity of $g$ and each derivative follows from that of $H$ and $D^{j} H$ : increments from $y$ to $y^{'}$ become the equal increments from $a + y$ to $a + y^{'}$ , and the derivative-value norms are unchanged. Thus $g$ is $C^{r}$ , including in the iterated operator norms.

3.1givenstep 1.1step 2.1

Apply step 2.1 with $H = h$ , $a = p$ , and $b = - p$ to obtain that $f$ is $C^{r}$ . For any $q \in U$ , choose an open neighbourhood $N \subseteq U$ on which $h$ has a $C^{r}$ inverse $k : h (N) \to N$ , with $h (N)$ open. Then $f$ maps $N - p$ bijectively onto $h (N) - p$ , with inverse $z \mapsto k (p + z) - p$ . Step 2.1 applied to $k$ proves that inverse is $C^{r}$ , and step 1.1 makes both domains open. This proves local diffeomorphism at every $q - p \in V$ . Also $f (0) = h (p) - p = 0$ .

4.1F1step 1.1step 3.1

The chain rule and the identity derivatives of translation give $D f (0) = I D h (p) I = D h (p)$ .

5.1step 4.1F3F4F5∎

Since the real operators are equal, their complexifications are equal by the formula in [F4]. Hence for every $λ \in C$ and integer $k \geq 1$ the powers $(D f (0)_{C} - λ I)^{k}$ and $(D h (p)_{C} - λ I)^{k}$ and their kernels agree. Thus the generalized eigenspaces, their conjugation-invariant sums, and the fixed real forms of those sums coincide. The fixed forms are regarded in the original real space: writing a tensor as $1 \otimes x + i \otimes y$ shows that being fixed by $σ$ forces $y = 0$ , since $v \mapsto 1 \otimes v$ is an embedding. This also verifies the stated real-subspace convention.

Depends on

Used by · 3

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.