Generating maps, invariant manifolds, conjugacy Marc Chaperon - PowerPoint PPT Presentation

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 ∗ R n = R n × R n ∗ = { ( q , p ) } h : T ∗ R n → T ∗ R n canonical: h ∗ ( p dq ) − p dq = d ϕ ( q , p ) . h deviates verticals, i.e. graph ( h ) ∋ ( q , p , Q , P ) �→ ( q , Q ) ∈ ( R n ) 2 diffeomorphism ⇒ h has a classical generating function a ( q , Q ) : graph ( h ) = { ( q , p , Q , P ) : P = ∂ Q a ( q , Q ) , p = − ∂ q a ( q , Q ) } Example. Rotation (Legendre map) ( q , p ) �→ ( − p ♯ , q ♭ ) : a ( q , Q ) = q · Q (scalar product). Composition: a j classical generating function of h j , 0 ≤ j ≤ N := � a j ( q j , q j + 1 ) ⇒ (Hamilton) a � � q 0 , q N + 1 ; ( q j ) 1 ≤ j ≤ N is a classical generating family (phase) of h = h N ◦ · · · ◦ h 0 : graph ( h ) set of ( q , p , Q , P ) such that for some v = ( q j ) 1 ≤ j ≤ N p = − ∂ q a ( q , Q ; v ) P = ∂ Q a ( q , Q ; v ) 0 = ∂ v a ( q , Q ; v ) . Marc Chaperon Institut de Mathématiques de Jussieu Université Paris 7 Generating maps, invariant manifolds, conjugacy

h : T ∗ R n → T ∗ R n canonical. graph ( h ) ∋ ( q , p , Q , P ) �→ ( Q , p ) ∈ T ∗ R n diffeomorphism ⇒ h has a generating function f ( Q , p ) : graph ( h ) = { ( q , p , Q , P ) : P = ∂ Q f ( Q , p ) , q = ∂ p f ( Q , p ) } Example. Identity map: f ( Q , p ) = pQ . Composition: f j generating function of h j , 0 ≤ j ≤ N N N � � � � ⇒ f f j ( Q j + 1 , p j ) − Q N + 1 , p 0 ; ( Q j , p j ) 1 ≤ j ≤ N := p j Q j 0 1 is a generating family (phase) of h = h N ◦ · · · ◦ h 0 : graph ( h ) set of ( q , p , Q , P ) such that ∃ v = ( Q j , p j ) 1 ≤ j ≤ N P = ∂ Q f ( Q , p ; v ) = ∂ p f ( Q , p ; v ) q 0 = ∂ v f ( 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 = E s ⊕ E u , AE s ⊂ E s , AE u ⊂ E u , spec A s ⊂ {| z | < 1 } , spec A u ⊂ {| z | > 1 } . | A s | < 1 < | A − 1 u | − 1 . ⇒ E = E s × E u , A = A s × A u , 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 ( ≃ | A s | ) < 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 ) } if h ( z ) ∈ Z , ¯ h ( z ) := ∅ otherwise: h = { ( x , y , x ′ , y ′ ) ∈ Z 2 : x ′ = F ( x , y ′ ) , y = G ( x , y ′ ) } . graph ¯ Orbit of length ℓ of ¯ h : ( z 0 , . . . , z ℓ ) ∈ Z ℓ + 1 with z j + 1 ∈ ¯ h ( z j ) . ℓ th iterate of ¯ h ℓ of Z into itself: h : correspondence ¯ ¯ h ℓ ( z ) set of endpoints z ℓ of orbits ( z 0 , . . . , z ℓ ) with z 0 = 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 . h ℓ has a generating map ( F 0 λµ < 1 ⇒ every ¯ ℓ , G ℓ 0 ) : Z → Z max { λ ℓ d ( x , x ′ ) , λ d ( y , y ′ ) } F 0 ℓ ( x , y ) , F 0 ℓ ( x ′ , y ′ ) � � d ≤ G ℓ 0 ( x , y ) , G ℓ 0 ( x ′ , y ′ ) max { µ d ( x , x ′ ) , µ ℓ d ( y , y ′ ) } . � � d ≤ For each ( x , y ) ∈ Z , one orbit ( z 0 , . . . , z ℓ ) of ¯ h such that x 0 = x and y ℓ = y if ( x j , y j ) := z j . THEOREM 2. - In addition Y bounded and µ < 1 ⇒ for each x ∈ X one orbit ( z ℓ ) ℓ ≥ 0 of ¯ h such that x 0 = x : ϕ ( x ) := y 0 (= G ℓ 0 ( x , y ℓ ) for all ℓ ) ; ϕ : X → Y ℓ →∞ G ℓ 0 ( x , y ′ ℓ ) uniformly in x and ( y ′ ℓ ) ℓ ≥ 0 ∈ Y N ϕ ( x ) = lim hence Lip ϕ ≤ µ and W s := graph ( ϕ ) = ¯ h − 1 ( W s ) W s ∋ z �→ ¯ h ( z ) ∩ W s map ¯ h s : W s → W s , Lip ¯ h s ≤ λ . Marc Chaperon Institut de Mathématiques de Jussieu Université Paris 7 Generating maps, invariant manifolds, conjugacy

A slightly unusual notion and a little lemma U 1 , . . . , U n , E metric, A : U = U 1 × · · · × U n → E . Definition. ( c 1 , . . . , c n ) ∈ Lip n ( A ) ⇐ ⇒ A ( u ) , A ( u ′ ) ≤ max { c 1 d ( u 1 , u ′ 1 ) , . . . , c n d ( u n , u ′ � � d n ) } . LEMMA. - Φ : U × E → E with ( c 1 , . . . , c n , κ ) ∈ Lip n + 1 Φ , E complete, κ < 1, A ( u ) fixed point of x �→ Φ( u , x ) ⇒ A : U → E satisfies ( c 1 , . . . , c n ) ∈ Lip n A . Proof. If x = A ( u ) , x ′ = A ( u ′ ) then d ( x , x ′ ) = d � Φ( u , x ) , Φ( u ′ , x ′ ) � ⇒ d ( x , x ′ ) ≤ max { c 1 d ( u 1 , u ′ 1 ) , . . . , c n d ( u n , u ′ n ) , κ d ( x , x ′ ) } ; r.-h. s. strictly κ d ( x , x ′ ) ⇒ 0 < ( 1 − κ ) d ( x , x ′ ) ≤ 0, hence A ( u ) , A ( u ′ ) = d ( x , x ′ ) ≤ max { c 1 d ( u 1 , u ′ 1 ) , . . . , c n d ( u n , u ′ � � d n ) } . Marc Chaperon Institut de Mathématiques de Jussieu Université Paris 7 Generating maps, invariant manifolds, conjugacy

Generating maps and composition HYPOTHESES. - X 0 , Y 0 , X 1 , Y 1 , X 2 , Y 2 metric, Z j = X j × Y j ¯ h j correspondence of Z j − 1 into Z j , j = 1 , 2 generating map ( F j , G j − 1 ) : X j − 1 × Y j → X j × Y j − 1 ( κ j , λ j ) ∈ Lip 2 F j , ( µ j − 1 , ν j − 1 ) ∈ Lip 2 G j − 1 , λ 1 µ 1 < 1 , Y 1 complete. COMPOSITION LEMMA. - Then ¯ h 2 ◦ ¯ h 1 has a generating map ( F , G ) : X 0 × Y 2 → X 2 × Y 0 , which satisfies Lip 2 F ∋ ( κ 2 κ 1 , max { κ 2 λ 1 ν 1 , λ 2 } ) Lip 2 G ∋ ( max { µ 0 , ν 0 κ 1 µ 1 } , ν 0 ν 1 ) . For each ( x , y ) ∈ X 0 × Y 2 , one ( z 0 , z 1 , z 2 ) such that x 0 = x , y 2 = y and z j ∈ ¯ h j ( z j − 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. x 0 = x , y 2 = y and z j ∈ ¯ h j ( z j − 1 ) , j = 1 , 2 write x 1 = F 1 ( x , y 1 ) , y 0 = G 0 ( x , y 1 ) , x 2 = F 2 ( x 1 , y ) , y 1 = G 1 ( x 1 , y ) . � � y 1 = G 1 F 1 ( x , y 1 ) , y =: Φ( x , y , y 1 ); (1) Lip 3 Φ ∋ ( µ 1 κ 1 , ν 1 , µ 1 λ 1 ) , µ 1 λ 1 < 1 ⇒ [lemma] (1) reads y 1 = A ( x , y ) (2) with Lip 2 A ∋ ( µ 1 κ 1 , ν 1 ) . Thus, our four equations are equivalent to (2) and � � � � y 0 = G ( x , y ) := G 0 x , A ( x , y ) , x 1 = F 1 x , A ( x , y ) � � � � x 2 = F ( x , y ) := F 2 F 1 x , A ( x , y ) , y , hence ∋ ( κ 2 κ 1 , max { κ 2 λ 1 ν 1 , λ 2 } ) Lip 2 F Lip 2 G ∋ ( max { µ 0 , ν 0 κ 1 µ 1 } , ν 0 ν 1 ) . Theorem 1 follows: induction step κ 1 = λ ℓ , λ 1 = κ 2 = λ 2 = λ , µ 1 = ν 1 = µ 0 = µ , ν 0 = µ ℓ . 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 ¯ h ℓ has a generating map ℓ , G ℓ 0 ) : Z → Z with ( λ ℓ , λ ) ∈ Lip 2 F 0 ℓ , ( µ, µ ℓ ) ∈ Lip 2 G ℓ ( F 0 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 ℓ ) ≤ µ ℓ diam Y G ℓ ℓ ) , G ℓ ≤ µ ℓ d ( y ′ � 0 ( x , y ′ 0 ( x , y ′′ � ℓ , y ′′ Indeed, d ℓ ) hence uniqueness of the limit if it exists and ≤ µ ℓ diam Y ℓ ) , G ℓ + 1 G ℓ � 0 ( x , y ′ ( x , y ′ � d ℓ + 1 ) 0 ℓ := y -component of the ℓ th term of the orbit of length [ y ′′ G ℓ ℓ + 1 defined by ( x , y ′ � 0 ( x , y ′ � ℓ + 1 ) ] ⇒ ℓ ) Cauchy, Lip ϕ ≤ µ . ◮ � � := orbit with x 0 ( ℓ ) = x , y ℓ ( ℓ ) = y ′ z 0 ( ℓ ) , . . . , z ℓ ( ℓ ) ℓ ⇒ lim ℓ →∞ z k ( ℓ ) = z k exists and depends only on x [functions A in the proof of the composition lemma] and ( z k ) is the only orbit with x 0 = x . Marc Chaperon Institut de Mathématiques de Jussieu Université Paris 7 Generating maps, invariant manifolds, conjugacy