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Generating maps, invariant manifolds, conjugacy

Marc Chaperon Institut de Mathématiques de Jussieu Université Paris 7 Symplectic Techniques in Dynamical Systems Madrid, November 13, 2013

Marc Chaperon Institut de Mathématiques de Jussieu Université Paris 7 Generating maps, invariant manifolds, conjugacy

Introduction: generating functions of canonical maps

T ∗Rn = Rn × Rn∗ = {(q, p)} h : T ∗Rn → T ∗Rn canonical: h∗(p dq) − p dq = dϕ(q, p). h deviates verticals, i.e. graph(h) ∋ (q, p, Q, P) → (q, Q) ∈ (Rn)2 diffeomorphism ⇒ h has a classical generating function a(q, Q): graph(h) = {(q, p, Q, P) : P = ∂Qa(q, Q), p = −∂qa(q, Q)}

• Example. Rotation (Legendre map) (q, p) → (−p♯, q♭):

a(q, Q) = q · Q (scalar product). Composition: aj classical generating function of hj, 0 ≤ j ≤ N ⇒ (Hamilton) a

• q0, qN+1; (qj)1≤j≤N
• := aj(qj, qj+1)

is a classical generating family (phase) of h = hN ◦ · · · ◦ h0: graph(h) set of (q, p, Q, P) such that for some v = (qj)1≤j≤N p = −∂qa(q, Q; v) P = ∂Qa(q, Q; v) = ∂va(q, Q; v).

Marc Chaperon Institut de Mathématiques de Jussieu Université Paris 7 Generating maps, invariant manifolds, conjugacy

h : T ∗Rn → T ∗Rn canonical. graph(h) ∋ (q, p, Q, P) → (Q, p) ∈ T ∗Rn diffeomorphism ⇒ h has a generating function f (Q, p): graph(h) = {(q, p, Q, P) : P = ∂Qf (Q, p), q = ∂pf (Q, p)}

• Example. Identity map: f (Q, p) = pQ.

Composition: fj generating function of hj, 0 ≤ j ≤ N ⇒ f

• QN+1, p0; (Qj, pj)1≤j≤N
• :=

N

• fj(Qj+1, pj) −

N

• 1

pjQj is a generating family (phase) of h = hN ◦ · · · ◦ h0: graph(h) set of (q, p, Q, P) such that ∃v = (Qj, pj)1≤j≤N P = ∂Qf (Q, p; v) q = ∂pf (Q, p; v) = ∂vf (Q, p; v).

• Remark. Applies to symplectic correspondences (relations) h.

Marc Chaperon Institut de Mathématiques de Jussieu Université Paris 7 Generating maps, invariant manifolds, conjugacy

A simple occurrence of generating maps

E Banach , h : (E, 0) → (E, 0) local C 1 map, A = Dh(0). 0 “hyperbolic” fixed point: E = Es ⊕ Eu, AEs ⊂ Es, AEu ⊂ Eu, spec As ⊂ {|z| < 1}, spec Au ⊂ {|z| > 1}. ⇒ E = Es × Eu, A = As × Au, |As| < 1 < |A−1

u |−1.

Invert the second component of h with respect to the second variable: If h = (f , g), then (x, y) →

• x, g(x, y)
• local diffeomorphism;

inverse (x, y′) →

• x, G(x, y′)
• .

Locally, graph h = {(x, y, x′, y′) : x′ = F(x, y′), y = G(x, y′)} with F(x, y′) := f

• x, G(x, y′)
• :

“(F, G) local generating map of h ”. Stable manifolds and the Perron-Irwin method, Ergodic Theory and Dynamical Systems 24 (2004), 1359–1394 The Lipschitzian core of some invariant manifold theorems,

• ibid. 28 (2008), 1419–1441.

Marc Chaperon Institut de Mathématiques de Jussieu Université Paris 7 Generating maps, invariant manifolds, conjugacy

Shut yourself in a small box Z r > 0 small, Z = X × Y =

• (x, y) : |(x, y)| ≤ r
• λ := Lip F|Z(≃ |As|) < 1, µ := Lip G|Z(≃ |A−1

u |) < 1

⇒ (F, G)(Z) ⊂ Z. (F, G) : Z → Z generating map of h|Z∩h−1(Z) viewed as the correspondence ¯ h of Z into itself (map Z → P(Z)) ¯ h(z) :=

• {h(z)}

if h(z) ∈ Z, ∅

• therwise:

graph ¯ h = {(x, y, x′, y′) ∈ Z 2 : x′ = F(x, y′), y = G(x, y′)}. Orbit of length ℓ of ¯ h: (z0, . . . , zℓ) ∈ Z ℓ+1 with zj+1 ∈ ¯ h(zj). ℓth iterate of ¯ h: correspondence ¯ hℓ of Z into itself: ¯ hℓ(z) set of endpoints zℓ of orbits (z0, . . . , zℓ) with z0 = z.

• R. McGehee, E. A. Sander. A new proof of the stable manifold
• theorem. Z. Angew. Math. Phys. 47, no. 4 (1996), 497–513.

Marc Chaperon Institut de Mathématiques de Jussieu Université Paris 7 Generating maps, invariant manifolds, conjugacy

THEOREM 1. - X, Y metric spaces, Y complete ¯ h correspondence of Z = X × Y into itself generating map (F 0

1 , G 1 0 ) : Z → Z, λ := Lip F 0 1 , µ := Lip G 1 0 .

λµ < 1 ⇒ every ¯ hℓ has a generating map (F 0

ℓ , G ℓ 0) : Z → Z

d

• F 0

ℓ (x, y), F 0 ℓ (x′, y′)

• ≤

max{λℓ d(x, x′), λ d(y, y′)} d

• G ℓ

0(x, y), G ℓ 0(x′, y′)

• ≤

max{µ d(x, x′), µℓd(y, y′)}. For each (x, y) ∈ Z, one orbit (z0, . . . , zℓ) of ¯ h such that x0 = x and yℓ = y if (xj, yj) := zj. THEOREM 2. - In addition Y bounded and µ < 1 ⇒ for each x ∈ X one orbit (zℓ)ℓ≥0 of ¯ h such that x0 = x: ϕ(x) := y0(= G ℓ

0(x, yℓ) for all ℓ); ϕ : X → Y

ϕ(x) = lim

ℓ→∞ G ℓ 0(x, y′ ℓ) uniformly in x and (y′ ℓ)ℓ≥0 ∈ Y N

hence Lip ϕ ≤ µ and Ws := graph(ϕ) = ¯ h−1(Ws) Ws ∋ z → ¯ h(z) ∩ Ws map ¯ hs : Ws → Ws, Lip ¯ hs ≤ λ.

Marc Chaperon Institut de Mathématiques de Jussieu Université Paris 7 Generating maps, invariant manifolds, conjugacy

A slightly unusual notion and a little lemma

U1, . . . , Un, E metric, A : U = U1 × · · · × Un → E.

• Definition. (c1, . . . , cn) ∈ Lipn(A) ⇐

⇒ d

• A(u), A(u′)
• ≤ max{c1d(u1, u′

1), . . . , cnd(un, u′ n)}.

• LEMMA. - Φ : U × E → E with (c1, . . . , cn, κ) ∈ Lipn+1 Φ,

E complete, κ < 1, A(u) fixed point of x → Φ(u, x) ⇒ A : U → E satisfies (c1, . . . , cn) ∈ Lipn A. Proof. If x = A(u), x′ = A(u′) then d(x, x′) = d

• Φ(u, x), Φ(u′, x′)
• ⇒ d(x, x′) ≤ max{c1d(u1, u′

1), . . . , cnd(un, u′ n), κd(x, x′)};

r.-h. s. strictly κd(x, x′) ⇒ 0 < (1 − κ)d(x, x′) ≤ 0, hence d

• A(u), A(u′)
• = d(x, x′) ≤ max{c1d(u1, u′

1), . . . , cnd(un, u′ n)}.

Marc Chaperon Institut de Mathématiques de Jussieu Université Paris 7 Generating maps, invariant manifolds, conjugacy

Generating maps and composition

• HYPOTHESES. - X0, Y0, X1, Y1, X2, Y2 metric, Zj = Xj × Yj

¯ hj correspondence of Zj−1 into Zj, j = 1, 2 generating map (Fj, Gj−1) : Xj−1 × Yj → Xj × Yj−1 (κj, λj) ∈ Lip2 Fj, (µj−1, νj−1) ∈ Lip2 Gj−1, λ1µ1 < 1, Y1 complete. COMPOSITION LEMMA. - Then ¯ h2 ◦ ¯ h1 has a generating map (F, G) : X0 × Y2 → X2 × Y0, which satisfies Lip2 F ∋ (κ2κ1, max{κ2λ1ν1, λ2}) Lip2 G ∋ (max{µ0, ν0κ1µ1}, ν0ν1). For each (x, y) ∈ X0 × Y2, one (z0, z1, z2) such that x0 = x, y2 = y and zj ∈ ¯ hj(zj−1), j = 1, 2. M.C. Invariant manifold theory via generating maps.

• C. R. Acad. Sci. Paris, Ser. I 346 (2008), 1175–1180.

Marc Chaperon Institut de Mathématiques de Jussieu Université Paris 7 Generating maps, invariant manifolds, conjugacy

• Proof. x0 = x, y2 = y and zj ∈ ¯

hj(zj−1), j = 1, 2 write x1 = F1(x, y1), y0 = G0(x, y1), x2 = F2(x1, y), y1 = G1(x1, y). y1 = G1

• F1(x, y1), y
• =: Φ(x, y, y1);

(1) Lip3 Φ ∋ (µ1κ1, ν1, µ1λ1), µ1λ1 < 1 ⇒ [lemma] (1) reads y1 = A(x, y) (2) with Lip2 A ∋ (µ1κ1, ν1). Thus, our four equations are equivalent to (2) and y0 = G(x, y) := G0

• x, A(x, y)
• ,

x1 = F1

• x, A(x, y)
• x2 = F(x, y)

:= F2

• F1
• x, A(x, y)
• , y
• , hence

Lip2 F ∋ (κ2κ1, max{κ2λ1ν1, λ2}) Lip2 G ∋ (max{µ0, ν0κ1µ1}, ν0ν1). Theorem 1 follows: induction step κ1 = λℓ, λ1 = κ2 = λ2 = λ, µ1 = ν1 = µ0 = µ, ν0 = µℓ.

Marc Chaperon Institut de Mathématiques de Jussieu Université Paris 7 Generating maps, invariant manifolds, conjugacy

Proof of Theorem 2. ¯ h correspondence of Z = X × Y into itself, generating map (F 0

1 , G 1 0 ), λ := Lip F 0 1 , µ := Lip G 1 0 ,

λµ < 1, Y complete ⇒ every ¯ hℓ has a generating map (F 0

ℓ , G ℓ 0) : Z → Z with (λℓ, λ) ∈ Lip2 F 0 ℓ , (µ, µℓ) ∈ Lip2 G ℓ 0.

Additional hypothesis. µ < 1, Y bounded.

◮ G ℓ 0(x, y′ ℓ) converges to ϕ(x), uniformly in x and (y′ ℓ) ∈ Y N

Indeed, d

• G ℓ

0(x, y′ ℓ), G ℓ 0(x, y′′ ℓ )

• ≤ µℓd(y′

ℓ, y′′ ℓ ) ≤ µℓ diam Y

hence uniqueness of the limit if it exists and d

• G ℓ

0(x, y′ ℓ), G ℓ+1

(x, y′

ℓ+1)

• ≤ µℓ diam Y

[y′′

ℓ := y-component of the ℓth term of the orbit of length

ℓ + 1 defined by (x, y′

ℓ+1)] ⇒

• G ℓ

0(x, y′ ℓ)

• Cauchy, Lip ϕ ≤ µ.

◮

z0(ℓ), . . . , zℓ(ℓ)

• := orbit with x0(ℓ) = x, yℓ(ℓ) = y′

ℓ

⇒ lim

ℓ→∞ zk(ℓ) = zk exists and depends only on x [functions A

in the proof of the composition lemma] and (zk) is the only

• rbit with x0 = x.

Marc Chaperon Institut de Mathématiques de Jussieu Université Paris 7 Generating maps, invariant manifolds, conjugacy