On the divergence of Birkhoff Normal Forms Rapha el KRIKORIAN CY - PowerPoint PPT Presentation

Invariant Tori Birkhoff Normal Forms are a key preliminary step to apply the KAM machinery and thus to prove the existence of (many) invariant tori and more generally of KAM tori. For ex. if f : p T d ˆ R d , T 0 q ý , Birkhoff Normal Forms 8 / 33

Invariant Tori Birkhoff Normal Forms are a key preliminary step to apply the KAM machinery and thus to prove the existence of (many) invariant tori and more generally of KAM tori. For ex. if f : p T d ˆ R d , T 0 q ý , § A (lagrangian) invariant torus of f : f -invariant set of the form Γ S S : T d Ñ R Γ S : “ tp θ, ∇ S p θ q , θ P T d u , Birkhoff Normal Forms 8 / 33

Invariant Tori Birkhoff Normal Forms are a key preliminary step to apply the KAM machinery and thus to prove the existence of (many) invariant tori and more generally of KAM tori. For ex. if f : p T d ˆ R d , T 0 q ý , § A (lagrangian) invariant torus of f : f -invariant set of the form Γ S S : T d Ñ R Γ S : “ tp θ, ∇ S p θ q , θ P T d u , § It is a KAM torus if the dynamics of f on Γ S is that of a (diophantine) translation. Birkhoff Normal Forms 8 / 33

KAM stability and BNF : smooth vs. real-analytic Birkhoff Normal Forms 9 / 33

KAM stability and BNF : smooth vs. real-analytic Theorem (K. A. M., R¨ ussmann, EFK) Assume that O “ t 0 u or T d ˆ t 0 u is a non resonant equilibrium of the smooth symplectic diffeom f : p R 2 d , 0 q ý or p T d ˆ R d , T 0 q ý . If B 8 “ BNF p f q is non-planar, then the origin is accumulated by a set of positive measure of invariant (KAM) tori. Birkhoff Normal Forms 9 / 33

KAM stability and BNF : smooth vs. real-analytic Theorem (K. A. M., R¨ ussmann, EFK) Assume that O “ t 0 u or T d ˆ t 0 u is a non resonant equilibrium of the smooth symplectic diffeom f : p R 2 d , 0 q ý or p T d ˆ R d , T 0 q ý . If B 8 “ BNF p f q is non-planar, then the origin is accumulated by a set of positive measure of invariant (KAM) tori. B 8 is non-planar or non-degenerate : if E γ P R d s.t. @ r , x ∇ B 8 p r q , γ y “ 0 . Birkhoff Normal Forms 9 / 33

KAM stability and BNF : smooth vs. real-analytic Theorem (K. A. M., R¨ ussmann, EFK) Assume that O “ t 0 u or T d ˆ t 0 u is a non resonant equilibrium of the smooth symplectic diffeom f : p R 2 d , 0 q ý or p T d ˆ R d , T 0 q ý . If B 8 “ BNF p f q is non-planar, then the origin is accumulated by a set of positive measure of invariant (KAM) tori. B 8 is non-planar or non-degenerate : if E γ P R d s.t. @ r , x ∇ B 8 p r q , γ y “ 0 . Theorem (R¨ ussmann) Assume f is real-analytic and ω 0 is Diophantine. If Φ B 8 “ Df p 0 q , then f is integrable : it is real-analytically conjugated to Df p 0 q in a neighborhood of the origin. Birkhoff Normal Forms 9 / 33

Divergence of BNF Birkhoff Normal Forms 10 / 33

Divergence of BNF ‚ Poincar´ e discovered that non integrability is generic : Even if f real analytic the conjugation relation Z 8 ˝ f ˝ Z ´ 1 8 “ Φ BNF p f q cannot hold with both Z 8 and BNF p f q converging. Birkhoff Normal Forms 10 / 33

Divergence of BNF ‚ Poincar´ e discovered that non integrability is generic : Even if f real analytic the conjugation relation Z 8 ˝ f ˝ Z ´ 1 8 “ Φ BNF p f q cannot hold with both Z 8 and BNF p f q converging. ‚ Siegel (1954) : the formal conjugacy Z 8 is in generically divergent. Birkhoff Normal Forms 10 / 33

Divergence of BNF ‚ Poincar´ e discovered that non integrability is generic : Even if f real analytic the conjugation relation Z 8 ˝ f ˝ Z ´ 1 8 “ Φ BNF p f q cannot hold with both Z 8 and BNF p f q converging. ‚ Siegel (1954) : the formal conjugacy Z 8 is in generically divergent. Eliasson’s Question. Are there examples of divergent BNF if f is analytic ? Birkhoff Normal Forms 10 / 33

Divergence of BNF ‚ Poincar´ e discovered that non integrability is generic : Even if f real analytic the conjugation relation Z 8 ˝ f ˝ Z ´ 1 8 “ Φ BNF p f q cannot hold with both Z 8 and BNF p f q converging. ‚ Siegel (1954) : the formal conjugacy Z 8 is in generically divergent. Eliasson’s Question. Are there examples of divergent BNF if f is analytic ? Birkhoff Normal Forms 10 / 33

Divergence of BNF erez-Marco’s Dichotomy (2003) : @ ω non-resonant § P´ fixed, either the BNF is always convergent or it prevalently diverges (Elliptic fixed point, hamiltonian case). Birkhoff Normal Forms 11 / 33

Divergence of BNF erez-Marco’s Dichotomy (2003) : @ ω non-resonant § P´ fixed, either the BNF is always convergent or it prevalently diverges (Elliptic fixed point, hamiltonian case). § Gong (2012), Yin (2015) : Examples of divergent BNF (for some ω Liouvillian, hamiltonian / diffeom.). Birkhoff Normal Forms 11 / 33

Divergence of BNF erez-Marco’s Dichotomy (2003) : @ ω non-resonant § P´ fixed, either the BNF is always convergent or it prevalently diverges (Elliptic fixed point, hamiltonian case). § Gong (2012), Yin (2015) : Examples of divergent BNF (for some ω Liouvillian, hamiltonian / diffeom.). Theorem (K) Let d ě 1 and ω 0 P R d non resonant. A generic (prevalent) real analytic symplectic diffeomorphism f : p R 2 d , 0 q ý , defined on max p| x | , | y |q ă 1 , with frequency vector ω 0 at 0, has a divergent BNF. Same result in (AA) case if ω 0 is diophantine. Birkhoff Normal Forms 11 / 33

Divergence of BNF erez-Marco’s Dichotomy (2003) : @ ω non-resonant § P´ fixed, either the BNF is always convergent or it prevalently diverges (Elliptic fixed point, hamiltonian case). § Gong (2012), Yin (2015) : Examples of divergent BNF (for some ω Liouvillian, hamiltonian / diffeom.). Theorem (K) Let d ě 1 and ω 0 P R d non resonant. A generic (prevalent) real analytic symplectic diffeomorphism f : p R 2 d , 0 q ý , defined on max p| x | , | y |q ă 1 , with frequency vector ω 0 at 0, has a divergent BNF. Same result in (AA) case if ω 0 is diophantine. § By Perez-Marco’s theorem : d “ 1 is enough. Birkhoff Normal Forms 11 / 33

Divergence of BNF erez-Marco’s Dichotomy (2003) : @ ω non-resonant § P´ fixed, either the BNF is always convergent or it prevalently diverges (Elliptic fixed point, hamiltonian case). § Gong (2012), Yin (2015) : Examples of divergent BNF (for some ω Liouvillian, hamiltonian / diffeom.). Theorem (K) Let d ě 1 and ω 0 P R d non resonant. A generic (prevalent) real analytic symplectic diffeomorphism f : p R 2 d , 0 q ý , defined on max p| x | , | y |q ă 1 , with frequency vector ω 0 at 0, has a divergent BNF. Same result in (AA) case if ω 0 is diophantine. § By Perez-Marco’s theorem : d “ 1 is enough. § Fayad (2019) : Explicit examples of divergent BNF in 3 , 4 degrees of freedom (hamiltonian case). Birkhoff Normal Forms 11 / 33

Divergence of BNF ussmann’s Theorem] If d “ 1 and Question. [Generalized R¨ ω is diophantine is it true that the convergence of the BNF implies that f is integrable in a neighborhood of 0 ? Birkhoff Normal Forms 12 / 33

Divergence of BNF ussmann’s Theorem] If d “ 1 and Question. [Generalized R¨ ω is diophantine is it true that the convergence of the BNF implies that f is integrable in a neighborhood of 0 ? Remark : B.Fayad constructed explicit counterexamples for hamiltonian system with 3,4 degrees of freedom. Birkhoff Normal Forms 12 / 33

Divergence of BNF ussmann’s Theorem] If d “ 1 and Question. [Generalized R¨ ω is diophantine is it true that the convergence of the BNF implies that f is integrable in a neighborhood of 0 ? Remark : B.Fayad constructed explicit counterexamples for hamiltonian system with 3,4 degrees of freedom. Question. Is a given f accumulated (in a strong analytic topology) by symplectic diffeomorphisms with convergent BNF ? Birkhoff Normal Forms 12 / 33

The mechanism for the divergence of the BNF Birkhoff Normal Forms 13 / 33

The mechanism for the divergence of the BNF In our appraoch, the divergence of the BNF comes from the following principle : Birkhoff Normal Forms 13 / 33

The mechanism for the divergence of the BNF In our appraoch, the divergence of the BNF comes from the following principle : The convergence of a formal object like the BNF has dynamical consequences. Birkhoff Normal Forms 13 / 33

The mechanism for the divergence of the BNF In our appraoch, the divergence of the BNF comes from the following principle : The convergence of a formal object like the BNF has dynamical consequences. Principle : If a real analytic symplectic diffeomorphism f : p R 2 , 0 q ý or p T ˆ R , T 0 q ý has a converging BNF, then it must have much more invariant tori than what a generic one has. Birkhoff Normal Forms 13 / 33

The mechanism for the divergence of the BNF In our appraoch, the divergence of the BNF comes from the following principle : The convergence of a formal object like the BNF has dynamical consequences. Principle : If a real analytic symplectic diffeomorphism f : p R 2 , 0 q ý or p T ˆ R , T 0 q ý has a converging BNF, then it must have much more invariant tori than what a generic one has. We illustrate this principle in the case where ω 0 is Diophantine with exponent τ : ´ ln min l P Z | k ω ´ l | τ p ω q “ lim sup ă 8 . ln k k Ñ8 Birkhoff Normal Forms 13 / 33

BNF For t ą 0 we define § L f p t q : the set of points in t r ă t u which are contained in an invariant circle Ă t r ă 2 t u ( r “ p 1 { 2 qp x 2 ` y 2 q ) Birkhoff Normal Forms 14 / 33

BNF For t ą 0 we define § L f p t q : the set of points in t r ă t u which are contained in an invariant circle Ă t r ă 2 t u ( r “ p 1 { 2 qp x 2 ` y 2 q ) § m f p t q “ Leb p T ˆs ´ t , t rz L f p t qq . Birkhoff Normal Forms 14 / 33

BNF For t ą 0 we define § L f p t q : the set of points in t r ă t u which are contained in an invariant circle Ă t r ă 2 t u ( r “ p 1 { 2 qp x 2 ` y 2 q ) § m f p t q “ Leb p T ˆs ´ t , t rz L f p t qq . Theorem (1) Let ω 0 be Diophantine. Assume that BNF p f qp r q is non-degenerate ( B 2 r BNF p f qp 0 q ą 0 ). Then, if BNF p f qp r q converges ˙ 2 β p ω 0 q ´ ˙ ˆ ˆ 1 m f p t q À exp ´ t 1 and β p ω 0 q “ 1 ` τ p ω 0 q . Birkhoff Normal Forms 14 / 33

Generic diffeomorphisms On the other hand Theorem (2) Let ω 0 be Diophantine. For a “generic” (prevalent) real analytic symplectic diffeomorphism f : p R 2 , 0 q ý or p T ˆ R , T 0 q ý with frequency ω 0 at 0 and non-degenerate BNF, there exists a sequence t j , lim t j “ 0 such that ˆ 1 ˙ β p ω 0 q ` ˙ ˆ m f p t j q Á exp ´ . t j Birkhoff Normal Forms 15 / 33

Summary Birkhoff Normal Forms Sketch of the proof of Theorem 1 Various Normal Forms Consequences of the convergence of the BNF Sketch of the proof of Theorem 1 16 / 33

We assume p T ˆ R , T 0 q ý f p θ, r q “ p θ ` 2 πω, r q ` p O p r q , O p r 2 qq , ω Diophantine “ Φ Ω ˝ f F , Ω p r q “ 2 πω 0 r ` b 2 r 2 , F p θ, r q “ O p r 3 q Sketch of the proof of Theorem 1 17 / 33

We assume p T ˆ R , T 0 q ý f p θ, r q “ p θ ` 2 πω, r q ` p O p r q , O p r 2 qq , ω Diophantine “ Φ Ω ˝ f F , Ω p r q “ 2 πω 0 r ` b 2 r 2 , F p θ, r q “ O p r 3 q b 2 ‰ 0, Ω P O σ p D p 0 , ¯ ρ qq , F P O σ p T h ˆ D p 0 , ¯ ρ qq : real-symmetric (wrt compl. conj.) holomorphic. Sketch of the proof of Theorem 1 17 / 33

We assume p T ˆ R , T 0 q ý f p θ, r q “ p θ ` 2 πω, r q ` p O p r q , O p r 2 qq , ω Diophantine “ Φ Ω ˝ f F , Ω p r q “ 2 πω 0 r ` b 2 r 2 , F p θ, r q “ O p r 3 q b 2 ‰ 0, Ω P O σ p D p 0 , ¯ ρ qq , F P O σ p T h ˆ D p 0 , ¯ ρ qq : real-symmetric (wrt compl. conj.) holomorphic. We can assume for some a ą 0 (apply BNF up to some order) ρ a . | F p θ, r q| : “ } F } h , ¯ ρ ď ¯ sup | ℑ θ |ă h , r P D p 0 , ¯ ρ q Fix 0 ă ρ ă ¯ ρ . Sketch of the proof of Theorem 1 17 / 33

Various types of approximate Normal Forms Sketch of the proof of Theorem 1 18 / 33

Various types of approximate Normal Forms We define various approximate Normal Forms : p g ˚ q ´ 1 ˝ Φ Ω p r q ˝ f F ˝ g ˚ “ Φ Ω ˚ p r q ˝ f F ˚ , F ˚ ! 1 } F ˚ } À exp p´p 1 { ρ q 2 β p ω 0 q´ q defined on various domains of the form W h , U “ T h ˆ U ˚ , U ˚ open set, ˚ “ BNF , KAM , HJ . Sketch of the proof of Theorem 1 18 / 33

Various types of approximate Normal Forms We define various approximate Normal Forms : p g ˚ q ´ 1 ˝ Φ Ω p r q ˝ f F ˝ g ˚ “ Φ Ω ˚ p r q ˝ f F ˚ , F ˚ ! 1 } F ˚ } À exp p´p 1 { ρ q 2 β p ω 0 q´ q defined on various domains of the form W h , U “ T h ˆ U ˚ , U ˚ open set, ˚ “ BNF , KAM , HJ . § r˚ “ BNF s : Approximate Birkhoff Normal Forms defined in T h ˆ D p 0 , ρ b q ( b “ τ ` 2). Sketch of the proof of Theorem 1 18 / 33

Various types of approximate Normal Forms We define various approximate Normal Forms : p g ˚ q ´ 1 ˝ Φ Ω p r q ˝ f F ˝ g ˚ “ Φ Ω ˚ p r q ˝ f F ˚ , F ˚ ! 1 } F ˚ } À exp p´p 1 { ρ q 2 β p ω 0 q´ q defined on various domains of the form W h , U “ T h ˆ U ˚ , U ˚ open set, ˚ “ BNF , KAM , HJ . § r˚ “ BNF s : Approximate Birkhoff Normal Forms defined in T h ˆ D p 0 , ρ b q ( b “ τ ` 2). § r˚ “ KAM s : Approximate KAM Normal Forms defined on domains T h ˆ U KAM where U KAM is a domain with holes. Sketch of the proof of Theorem 1 18 / 33

Various types of approximate Normal Forms We define various approximate Normal Forms : p g ˚ q ´ 1 ˝ Φ Ω p r q ˝ f F ˝ g ˚ “ Φ Ω ˚ p r q ˝ f F ˚ , F ˚ ! 1 } F ˚ } À exp p´p 1 { ρ q 2 β p ω 0 q´ q defined on various domains of the form W h , U “ T h ˆ U ˚ , U ˚ open set, ˚ “ BNF , KAM , HJ . § r˚ “ BNF s : Approximate Birkhoff Normal Forms defined in T h ˆ D p 0 , ρ b q ( b “ τ ` 2). § r˚ “ KAM s : Approximate KAM Normal Forms defined on domains T h ˆ U KAM where U KAM is a domain with holes. § r˚ “ HJ s : For each hole D of U KAM we find disks D Ą D , ˇ ˆ D Ă ˆ D and an approximate Hamilton-Jacobi Normal Form in T h ˆ p ˆ D z ˇ D q . Sketch of the proof of Theorem 1 18 / 33

Various types of approximate Normal Forms We define various approximate Normal Forms : p g ˚ q ´ 1 ˝ Φ Ω p r q ˝ f F ˝ g ˚ “ Φ Ω ˚ p r q ˝ f F ˚ , F ˚ ! 1 } F ˚ } À exp p´p 1 { ρ q 2 β p ω 0 q´ q defined on various domains of the form W h , U “ T h ˆ U ˚ , U ˚ open set, ˚ “ BNF , KAM , HJ . § r˚ “ BNF s : Approximate Birkhoff Normal Forms defined in T h ˆ D p 0 , ρ b q ( b “ τ ` 2). § r˚ “ KAM s : Approximate KAM Normal Forms defined on domains T h ˆ U KAM where U KAM is a domain with holes. § r˚ “ HJ s : For each hole D of U KAM we find disks D Ą D , ˇ ˆ D Ă ˆ D and an approximate Hamilton-Jacobi Normal Form in T h ˆ p ˆ D z ˇ D q . KAM overlaps with BNF and HJ. Sketch of the proof of Theorem 1 18 / 33

The KAM and BNF normal forms are constructed by successive conjugations on smaller and smaller complex domains T h i ˆ U i f Y i ˝ Φ Ω i ˝ f F i ˝ f ´ 1 “ Φ Ω i ` 1 ˝ f F i ` 1 F i ` 1 “ O 2 p F i q Y i ż Ω i ` 1 p r q “ Ω i p r q ` p 2 π q ´ 1 F i p θ, r q d θ. T Sketch of the proof of Theorem 1 19 / 33

The KAM and BNF normal forms are constructed by successive conjugations on smaller and smaller complex domains T h i ˆ U i f Y i ˝ Φ Ω i ˝ f F i ˝ f ´ 1 “ Φ Ω i ` 1 ˝ f F i ` 1 F i ` 1 “ O 2 p F i q Y i ż Ω i ` 1 p r q “ Ω i p r q ` p 2 π q ´ 1 F i p θ, r q d θ. T by solving the (truncated) linearized equation ż F i ´ p 2 π q ´ 1 F i p θ, ¨q d θ “ Y i ˝ Φ Ω i ´ Y i (4) T Sketch of the proof of Theorem 1 19 / 33

The KAM and BNF normal forms are constructed by successive conjugations on smaller and smaller complex domains T h i ˆ U i f Y i ˝ Φ Ω i ˝ f F i ˝ f ´ 1 “ Φ Ω i ` 1 ˝ f F i ` 1 F i ` 1 “ O 2 p F i q Y i ż Ω i ` 1 p r q “ Ω i p r q ` p 2 π q ´ 1 F i p θ, r q d θ. T by solving the (truncated) linearized equation ż F i ´ p 2 π q ´ 1 F i p θ, ¨q d θ “ Y i ˝ Φ Ω i ´ Y i (4) T Sketch of the proof of Theorem 1 19 / 33

The KAM and BNF normal forms are constructed by successive conjugations on smaller and smaller complex domains T h i ˆ U i f Y i ˝ Φ Ω i ˝ f F i ˝ f ´ 1 “ Φ Ω i ` 1 ˝ f F i ` 1 F i ` 1 “ O 2 p F i q Y i ż Ω i ` 1 p r q “ Ω i p r q ` p 2 π q ´ 1 F i p θ, r q d θ. T by solving the (truncated) linearized equation ż F i ´ p 2 π q ´ 1 F i p θ, ¨q d θ “ Y i ˝ Φ Ω i ´ Y i (4) T § For KAM : avoid resonances when solving (4) | k ∇ Ω i p r q ´ 2 π l | ě K ´ 1 , @ 0 ă | k | ă N i (5) i hence U i ` 1 “ U i z Ť disks , N 2 i disks radii K ´ 1 i Sketch of the proof of Theorem 1 19 / 33

The KAM and BNF normal forms are constructed by successive conjugations on smaller and smaller complex domains T h i ˆ U i f Y i ˝ Φ Ω i ˝ f F i ˝ f ´ 1 “ Φ Ω i ` 1 ˝ f F i ` 1 F i ` 1 “ O 2 p F i q Y i ż Ω i ` 1 p r q “ Ω i p r q ` p 2 π q ´ 1 F i p θ, r q d θ. T by solving the (truncated) linearized equation ż F i ´ p 2 π q ´ 1 F i p θ, ¨q d θ “ Y i ˝ Φ Ω i ´ Y i (4) T § For KAM : avoid resonances when solving (4) | k ∇ Ω i p r q ´ 2 π l | ě K ´ 1 , @ 0 ă | k | ă N i (5) i hence U i ` 1 “ U i z Ť disks , N 2 i disks radii K ´ 1 i § For (approx.) BNF U i are smaller and smaller disks centered at 0 (essentially no resonances). Sketch of the proof of Theorem 1 19 / 33

b KAM and BNF Normal Forms D p 0 , ρ q Ω KAM Ω BNF on D p 0 , ρ b q 0 R -axis D hole of KAM Ξ “ BNF Figure : KAM and BNF Normal Forms Sketch of the proof of Theorem 1 20 / 33

Hamilton-Jacobi NF If one faces a resonance at the i -th step of KAM : (5) fails p 2 π q ´ 1 B Ω p c q “ l { k , | k | ă N i . Sketch of the proof of Theorem 1 21 / 33

Hamilton-Jacobi NF If one faces a resonance at the i -th step of KAM : (5) fails p 2 π q ´ 1 B Ω p c q “ l { k , | k | ă N i . § Make a resonant NF (similar to BNF) to eliminate harmonics R k Z : can be done on T h { 3 ˆ D p c , ˆ K ´ 1 q i ˆ K ´ 1 “ N ´ ln N i . (Compare to K ´ 1 “ exp p´ N i {p ln N i q a q ). i i i Sketch of the proof of Theorem 1 21 / 33

Hamilton-Jacobi NF If one faces a resonance at the i -th step of KAM : (5) fails p 2 π q ´ 1 B Ω p c q “ l { k , | k | ă N i . § Make a resonant NF (similar to BNF) to eliminate harmonics R k Z : can be done on T h { 3 ˆ D p c , ˆ K ´ 1 q i ˆ K ´ 1 “ N ´ ln N i . (Compare to K ´ 1 “ exp p´ N i {p ln N i q a q ). i i i § Rescale (covering) and get a system very close to a hamiltonian in T kh { 3 ˆ D p 0 , k ˆ K ´ 1 q : Pendulum like. i Sketch of the proof of Theorem 1 21 / 33

Hamilton-Jacobi NF If one faces a resonance at the i -th step of KAM : (5) fails p 2 π q ´ 1 B Ω p c q “ l { k , | k | ă N i . § Make a resonant NF (similar to BNF) to eliminate harmonics R k Z : can be done on T h { 3 ˆ D p c , ˆ K ´ 1 q i ˆ K ´ 1 “ N ´ ln N i . (Compare to K ´ 1 “ exp p´ N i {p ln N i q a q ). i i i § Rescale (covering) and get a system very close to a hamiltonian in T kh { 3 ˆ D p 0 , k ˆ K ´ 1 q : Pendulum like. i § This pendulum on the cylinder is integrable outside the eye : perform Hamilton-Jacobi to this vector field. Sketch of the proof of Theorem 1 21 / 33

Hamilton-Jacobi NF If one faces a resonance at the i -th step of KAM : (5) fails p 2 π q ´ 1 B Ω p c q “ l { k , | k | ă N i . § Make a resonant NF (similar to BNF) to eliminate harmonics R k Z : can be done on T h { 3 ˆ D p c , ˆ K ´ 1 q i ˆ K ´ 1 “ N ´ ln N i . (Compare to K ´ 1 “ exp p´ N i {p ln N i q a q ). i i i § Rescale (covering) and get a system very close to a hamiltonian in T kh { 3 ˆ D p 0 , k ˆ K ´ 1 q : Pendulum like. i § This pendulum on the cylinder is integrable outside the eye : perform Hamilton-Jacobi to this vector field. § Come back. Sketch of the proof of Theorem 1 21 / 33

Hamilton-Jacobi NF One gets a NF defined on T h { 20 ˆ ˆ D z ˇ D D “ D p c , ˆ ˆ ˇ K ´ 1 q , D corresponds to the eye i Sketch of the proof of Theorem 1 22 / 33

Hamilton-Jacobi NF One gets a NF defined on T h { 20 ˆ ˆ D z ˇ D D “ D p c , ˆ ˆ ˇ K ´ 1 q , D corresponds to the eye i g ´ 1 ˝ Φ Ω i ˝ f F i ˝ g i “ Φ Ω HJ D ˝ f F HJ D . i Sketch of the proof of Theorem 1 22 / 33

b The various Normal Forms D p 0 , ρ q Ω KAM Ω BNF on D p 0 , ρ b q 0 R -axis D hole of KAM Ξ “ BNF Figure : KAM and BNF Normal Forms Sketch of the proof of Theorem 1 23 / 33

b The various Normal Forms D p 0 , ρ q Ω KAM D on ˆ D z ˇ Ω BNF on Ω HJ D D z ˇ ˆ D p 0 , ρ b q 0 R -axis D hole of KAM Ξ “ BNF Figure : The various Normal Forms Sketch of the proof of Theorem 1 24 / 33

The Extension Principle A priori one might think there is no gain in doing this. Sketch of the proof of Theorem 1 25 / 33

The Extension Principle A priori one might think there is no gain in doing this. Interest : Sketch of the proof of Theorem 1 25 / 33

The Extension Principle A priori one might think there is no gain in doing this. Interest : One knows how this Hamilton-Jacobi NF is constructed. In particular it satisfies the Sketch of the proof of Theorem 1 25 / 33

The Extension Principle A priori one might think there is no gain in doing this. Interest : One knows how this Hamilton-Jacobi NF is constructed. In particular it satisfies the Extension Principle If there exists a holomorphic function Ξ P O p ˆ D q such that } Ω HJ D ´ Ξ } p 4 { 5 q ˆ D À ν D zp 1 { 5 q ˆ then radius p ˇ D q À ν 1 { 43 . Sketch of the proof of Theorem 1 25 / 33

The Extension Principle A priori one might think there is no gain in doing this. Interest : One knows how this Hamilton-Jacobi NF is constructed. In particular it satisfies the Extension Principle If there exists a holomorphic function Ξ P O p ˆ D q such that } Ω HJ D ´ Ξ } p 4 { 5 q ˆ D À ν D zp 1 { 5 q ˆ then radius p ˇ D q À ν 1 { 43 . Amounts to Residue Principle applied to p z 2 ` a 2 q 1 { 2 . Sketch of the proof of Theorem 1 25 / 33

Matching of these Normal Forms These Normal Forms almost coincide on their domain of definitions : Sketch of the proof of Theorem 1 26 / 33

Matching of these Normal Forms These Normal Forms almost coincide on their domain of definitions : Matching Principle If on an annulus like domain g ´ 1 ˝ Φ Ω p r q ˝ f F ˝ g j “ Φ Ω p r q i ˝ f F j , j “ 1 , 2 j where F j are small then ∇ Ω 1 and ∇ Ω 2 coincide with good approximation. Sketch of the proof of Theorem 1 26 / 33

Matching of these Normal Forms These Normal Forms almost coincide on their domain of definitions : Matching Principle If on an annulus like domain g ´ 1 ˝ Φ Ω p r q ˝ f F ˝ g j “ Φ Ω p r q i ˝ f F j , j “ 1 , 2 j where F j are small then ∇ Ω 1 and ∇ Ω 2 coincide with good approximation. § Ω BNF « Ω KAM on D p 0 , ρ b q . Sketch of the proof of Theorem 1 26 / 33

Matching of these Normal Forms These Normal Forms almost coincide on their domain of definitions : Matching Principle If on an annulus like domain g ´ 1 ˝ Φ Ω p r q ˝ f F ˝ g j “ Φ Ω p r q i ˝ f F j , j “ 1 , 2 j where F j are small then ∇ Ω 1 and ∇ Ω 2 coincide with good approximation. § Ω BNF « Ω KAM on D p 0 , ρ b q . § Ω KAM « Ω HJ D on ˆ D z ˇ D D z ˇ ˆ Sketch of the proof of Theorem 1 26 / 33

Matching of these Normal Forms These Normal Forms almost coincide on their domain of definitions : Matching Principle If on an annulus like domain g ´ 1 ˝ Φ Ω p r q ˝ f F ˝ g j “ Φ Ω p r q i ˝ f F j , j “ 1 , 2 j where F j are small then ∇ Ω 1 and ∇ Ω 2 coincide with good approximation. § Ω BNF « Ω KAM on D p 0 , ρ b q . § Ω KAM « Ω HJ D on ˆ D z ˇ D D z ˇ ˆ Furthermore if the BNF converges on D p 0 , 1 q : Ω BNF « Ξ “ BNF , on D p 0 , ρ b q . Sketch of the proof of Theorem 1 26 / 33

Convergence of the BNF implies Ω KAM « BNF Sketch of the proof of Theorem 1 27 / 33

Convergence of the BNF implies Ω KAM « BNF No-Screening Principle : Let U be a disk with holes, ∆ Ă U a disk and ϕ : U Ñ C a holomorphic function which is small on ∆. Then, ϕ has to be small on (a smaller domain contained in) U provided one controls the Green function of U . Sketch of the proof of Theorem 1 27 / 33

Convergence of the BNF implies Ω KAM « BNF No-Screening Principle : Let U be a disk with holes, ∆ Ă U a disk and ϕ : U Ñ C a holomorphic function which is small on ∆. Then, ϕ has to be small on (a smaller domain contained in) U provided one controls the Green function of U . If the BNF converges this implies that Sketch of the proof of Theorem 1 27 / 33

Convergence of the BNF implies Ω KAM « BNF No-Screening Principle : Let U be a disk with holes, ∆ Ă U a disk and ϕ : U Ñ C a holomorphic function which is small on ∆. Then, ϕ has to be small on (a smaller domain contained in) U provided one controls the Green function of U . If the BNF converges this implies that Ω KAM « Ξ “ BNF on U KAM Sketch of the proof of Theorem 1 27 / 33

Convergence of the BNF implies Ω KAM « BNF No-Screening Principle : Let U be a disk with holes, ∆ Ă U a disk and ϕ : U Ñ C a holomorphic function which is small on ∆. Then, ϕ has to be small on (a smaller domain contained in) U provided one controls the Green function of U . If the BNF converges this implies that Ω KAM « Ξ “ BNF on U KAM and thus from the Matching Property on ˆ D z ˇ Ω HJ D « Ξ “ BNF D . D z ˇ ˆ Sketch of the proof of Theorem 1 27 / 33