LemmaStatement: AI-generatedProof: AI-generated

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Transport local invariance and forward trapping to a fixed point

Statement

Let $m \geq 0$ and $r \geq 1$ be integers, with $r$ finite. Let $U \subseteq R^{m}$ be open, let $p \in U$ , and let $h : U \to R^{m}$ be a $C^{r}$ local diffeomorphism with $h (p) = p$ . Here $C^{r}$ uses continuous iterated Fréchet derivatives through order $r$ , and local diffeomorphism means that around each point $h$ has a $C^{r}$ inverse onto an open image. Define $f : U - p \to R^{m}$ by $f (y) = h (p + y) - p$ . For either local map, an iterate is defined only when all preceding points lie in its domain; $f^{0}$ and $h^{0}$ are identities.

Let $0 \leq d \leq m$ be integers, let $s \in {1, 2, \dots} \cup {\infty}$ , and let $W$ be a $C^{s}$ embedded $d$ -disc through zero. An embedded disc here is an image $k (E)$ of an injective $C^{s}$ map $k : E \to R^{m}$ from a nonempty open ball $E \subseteq R^{d}$ , such that $k$ is a homeomorphism onto its image with the subspace topology and $D k$ is injective everywhere. Use the componentwise $C^{s}$ and zero-dimensional conventions of the translated-disc lemma.

Let $B \subseteq U - p$ be an open ball about zero for some norm. Suppose $f (W \cap (U - p)) \cap B \subseteq W$ , and suppose every $y \in U - p$ for which all $f^{n} (y)$ are defined and lie in $B$ for $n \geq 0$ belongs to $W$ . Then $W_{p} = p + W$ is a $C^{s}$ embedded $d$ -disc through $p$ with $T_{p} W_{p} = T_{0} W$ under the Euclidean tangent-space identifications, $B_{p} = p + B$ is a Euclidean open neighbourhood of $p$ , $h (W_{p} \cap U) \cap B_{p} \subseteq W_{p}$ , and every $x \in U$ for which all $h^{n} (x)$ are defined and lie in $B_{p}$ for $n \geq 0$ belongs to $W_{p}$ .

Facts & Assumptions

[F1]

Under the stated open-domain, fixed-point and finite-regularity local-diffeomorphism hypotheses, translation supplies the local conjugacy Translate a local diffeomorphism and its spectral data to the origin and identities for every defined iterate Translate every finite forward orbit without assuming global invertibility.

[F2]

The embedded-disc, componentwise regularity and zero-dimensional conventions, and preservation of discs and tangent spaces under translation, are supplied by Translation preserves an embedded disc and its tangent space. Norm balls are Euclidean neighbourhoods by Adapted norm balls are ordinary Euclidean neighbourhoods.

Proof

Given: The map, open domain, fixed point, finite integer map regularity, defined-iterate convention, disc and its independent regularity and embedding convention, ball, and invariance and trapping hypotheses in the Statement.

Goal: Transport all these conclusions to the fixed point $p$ .

1.1givenF2

Choose the admissible parametrization $k : E \to W$ and $a_{0} \in E$ with $k (a_{0}) = 0$ . Put $D = E - a_{0}$ and $j (a) = k (a + a_{0})$ . Then $D$ is an open ball containing zero, $j (0) = 0$ and $j (D) = W$ . Parameter translation and its inverse preserve distances and hence are homeomorphisms; composing them with $k$ and its inverse shows that $j$ is a homeomorphism onto $W$ . It is injective. In positive source and target dimensions, coordinate difference quotients give $\partial^{α} j (a) = \partial^{α} k (a + a_{0})$ for every nonempty derivative word $α$ of required order, so these derivatives are continuous and $j$ is $C^{s}$ . The differentiability remainder likewise gives $D j (a) = D k (a + a_{0})$ , which is injective. In zero source or target dimension, the same regularity and derivative conclusions follow from the zero-dimensional convention. Thus $j$ is an admissible parametrization of the entire disc satisfying the centered-parametrization hypotheses of the translated-disc result.

1.2givenF1

If $z \in h (W_{p} \cap U) \cap B_{p}$ , write $z = h (p + w)$ with $w \in W \cap (U - p)$ . Then $z - p = f (w) \in B$ , so $z - p \in f (W \cap (U - p)) \cap B \subseteq W$ . Hence $z \in W_{p}$ .

2.1givenF2step 1.1

Apply the translated-disc result in [F2] to $j$ and $D$ from step 1.1. It gives that $W_{p}$ is a $C^{s}$ embedded $d$ -disc through $p$ with $T_{p} W_{p} = T_{0} W$ . The norm-ball result in [F2] gives that $B_{p}$ is a Euclidean open neighbourhood of $p$ .

3.1givenF1∎

Let $x \in U$ have all nonnegative $h$ -iterates defined and in $B_{p}$ . The defined-iterate identities in [F1] give that all $f^{n} (x - p)$ are defined and lie in $B$ for every $n \geq 0$ . The trapping hypothesis gives $x - p \in W$ , so $x \in W_{p}$ , as required.

Depends on

Used by · 1

Local center-stable trapping disc for a time map

Dependency tree · 2 levels

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