On the divergence of Birkhoff Normal Forms Rapha el KRIKORIAN CY - - PowerPoint PPT Presentation
On the divergence of Birkhoff Normal Forms Rapha el KRIKORIAN CY Cergy Paris Universit e July 7th, 2020 Lyapunov Exponents Lisbon (on the web) 1 / 33 Summary Birkhoff Normal Forms Elliptic equilibria Birkhoff Normal Forms KAM and
Rapha¨ el KRIKORIAN
CY Cergy Paris Universit´ e
July 7th, 2020 Lyapunov Exponents Lisbon (on the web)
1 / 33
Birkhoff Normal Forms Elliptic equilibria Birkhoff Normal Forms KAM and BNF : smooth vs. real-analytic Divergence of BNF, Theorems 1 and 2 Sketch of the proof of Theorem 1
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Td “ Rd{p2πZqd.
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Td “ Rd{p2πZqd. f diffeomorphism C ω, symplectic
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Td “ Rd{p2πZqd. f diffeomorphism C ω, symplectic (CC)-case :
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Td “ Rd{p2πZqd. f diffeomorphism C ω, symplectic (CC)-case : f : pR2d, 0q ý, d ě 1 f “ Df p0q ˝ pid ` O2px, yqq (1) Df p0q symplectic rotation with e.v. e˘2πiωj, 1 ď j ď d
Birkhoff Normal Forms 3 / 33
Td “ Rd{p2πZqd. f diffeomorphism C ω, symplectic (CC)-case : f : pR2d, 0q ý, d ě 1 f “ Df p0q ˝ pid ` O2px, yqq (1) Df p0q symplectic rotation with e.v. e˘2πiωj, 1 ď j ď d (AA)-case :
Birkhoff Normal Forms 3 / 33
Td “ Rd{p2πZqd. f diffeomorphism C ω, symplectic (CC)-case : f : pR2d, 0q ý, d ě 1 f “ Df p0q ˝ pid ` O2px, yqq (1) Df p0q symplectic rotation with e.v. e˘2πiωj, 1 ď j ď d (AA)-case : f : pTd ˆ Rd, T0q ý, T0 :“ Td ˆ t0u, d ě 1, f pθ, rq “ pθ ` 2πω, rq ` pOprq, Opr 2qq, ω P Rd (2)
Birkhoff Normal Forms 3 / 33
Td “ Rd{p2πZqd. f diffeomorphism C ω, symplectic (CC)-case : f : pR2d, 0q ý, d ě 1 f “ Df p0q ˝ pid ` O2px, yqq (1) Df p0q symplectic rotation with e.v. e˘2πiωj, 1 ď j ď d (AA)-case : f : pTd ˆ Rd, T0q ý, T0 :“ Td ˆ t0u, d ě 1, f pθ, rq “ pθ ` 2πω, rq ` pOprq, Opr 2qq, ω P Rd (2) ω “ pω1, . . . , ωdq is the frequency vector (at the origin). Notation rj “ p1{2qpx2
j ` y 2 j q, r “ pr1, . . . , rdq.
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One can write f “ ΦΩ ˝ fF, Ωprq “ 2πxω, ry (3)
Birkhoff Normal Forms 4 / 33
One can write f “ ΦΩ ˝ fF, Ωprq “ 2πxω, ry (3)
Birkhoff Normal Forms 4 / 33
One can write f “ ΦΩ ˝ fF, Ωprq “ 2πxω, ry (3)
§ ΦΩ : integrable model (associated to Ω : r ÞÑ Ωprq).
§ (CC) case : ΦΩ is a generalized symplectic rotation
ΦΩpx, yq “ p˜ x, ˜ yq ð ñ # ˜ xj ` i˜ yj “ e∇Ωprqpxj ` iyjq @ 1 ď j ď d ;
§ (AA)-case
ΦΩpθ, rq “ pθ ` ∇Ωprq, rq
Birkhoff Normal Forms 4 / 33
One can write f “ ΦΩ ˝ fF, Ωprq “ 2πxω, ry (3)
§ ΦΩ : integrable model (associated to Ω : r ÞÑ Ωprq).
§ (CC) case : ΦΩ is a generalized symplectic rotation
ΦΩpx, yq “ p˜ x, ˜ yq ð ñ # ˜ xj ` i˜ yj “ e∇Ωprqpxj ` iyjq @ 1 ď j ď d ;
§ (AA)-case
ΦΩpθ, rq “ pθ ` ∇Ωprq, rq
§ fF perturbation : exact sympl. diffeom. assoc. to
F : R2d Ñ R of F : Td ˆ Rd Ñ R.
Birkhoff Normal Forms 4 / 33
One can write f “ ΦΩ ˝ fF, Ωprq “ 2πxω, ry (3)
§ ΦΩ : integrable model (associated to Ω : r ÞÑ Ωprq).
§ (CC) case : ΦΩ is a generalized symplectic rotation
ΦΩpx, yq “ p˜ x, ˜ yq ð ñ # ˜ xj ` i˜ yj “ e∇Ωprqpxj ` iyjq @ 1 ď j ď d ;
§ (AA)-case
ΦΩpθ, rq “ pθ ` ∇Ωprq, rq
§ fF perturbation : exact sympl. diffeom. assoc. to
F : R2d Ñ R of F : Td ˆ Rd Ñ R. For example in the (AA) case fFpθ, rq “ pϕ, Rq ð ñ # r “ R ` BθFpθ, Rq ϕ “ θ ` BRFpθ, Rq
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ω :“ pω1, . . . , ωdq : frequency vector at the origin.
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ω :“ pω1, . . . , ωdq : frequency vector at the origin.
§ We say ω non resonant if :
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ω :“ pω1, . . . , ωdq : frequency vector at the origin.
§ We say ω non resonant if :
# k0 ` řd
j“1 kjωj “ 0
k0, k1, . . . , kd P Z ù ñ k0 “ k1 “ . . . “ kd “ 0.
Birkhoff Normal Forms 6 / 33
ω :“ pω1, . . . , ωdq : frequency vector at the origin.
§ We say ω non resonant if :
# k0 ` řd
j“1 kjωj “ 0
k0, k1, . . . , kd P Z ù ñ k0 “ k1 “ . . . “ kd “ 0.
§ If ω0 is non resonnant we say that 0 or T ˆ t0u is a non
resonant equilibrium of f .
Birkhoff Normal Forms 6 / 33
ω :“ pω1, . . . , ωdq : frequency vector at the origin.
§ We say ω non resonant if :
# k0 ` řd
j“1 kjωj “ 0
k0, k1, . . . , kd P Z ù ñ k0 “ k1 “ . . . “ kd “ 0.
§ If ω0 is non resonnant we say that 0 or T ˆ t0u is a non
resonant equilibrium of f .
§ ω0 Diophantine if
@ k P Zdzt0u, min
lPZ |xk, ωy ´ l| ě
γ |k|τ pτ ě 1q
Birkhoff Normal Forms 6 / 33
ω :“ pω1, . . . , ωdq : frequency vector at the origin.
§ We say ω non resonant if :
# k0 ` řd
j“1 kjωj “ 0
k0, k1, . . . , kd P Z ù ñ k0 “ k1 “ . . . “ kd “ 0.
§ If ω0 is non resonnant we say that 0 or T ˆ t0u is a non
resonant equilibrium of f .
§ ω0 Diophantine if
@ k P Zdzt0u, min
lPZ |xk, ωy ´ l| ě
γ |k|τ pτ ě 1q
§ (CC) case : ω0 non-resonant.
Birkhoff Normal Forms 6 / 33
ω :“ pω1, . . . , ωdq : frequency vector at the origin.
§ We say ω non resonant if :
# k0 ` řd
j“1 kjωj “ 0
k0, k1, . . . , kd P Z ù ñ k0 “ k1 “ . . . “ kd “ 0.
§ If ω0 is non resonnant we say that 0 or T ˆ t0u is a non
resonant equilibrium of f .
§ ω0 Diophantine if
@ k P Zdzt0u, min
lPZ |xk, ωy ´ l| ě
γ |k|τ pτ ě 1q
§ (CC) case : ω0 non-resonant. § (AA) case : ω0 Diophantine.
Birkhoff Normal Forms 6 / 33
Birkhoff Normal Forms 7 / 33
The Birkhoff Normal Form Theorem : Formal Integrability.
Birkhoff Normal Forms 7 / 33
The Birkhoff Normal Form Theorem : Formal Integrability. Let f : pR2d, 0q ý or pTd ˆ Rd, T0q ý be a smooth sympl.
Birkhoff Normal Forms 7 / 33
The Birkhoff Normal Form Theorem : Formal Integrability. Let f : pR2d, 0q ý or pTd ˆ Rd, T0q ý be a smooth sympl.
§ @N ě 1, D ZN (exact) sympl., BN P RNrr1, . . . , rds
ZN ˝ f ˝ Z ´1
N
“ ΦBN ` ON`1prq.
Birkhoff Normal Forms 7 / 33
The Birkhoff Normal Form Theorem : Formal Integrability. Let f : pR2d, 0q ý or pTd ˆ Rd, T0q ý be a smooth sympl.
§ @N ě 1, D ZN (exact) sympl., BN P RNrr1, . . . , rds
ZN ˝ f ˝ Z ´1
N
“ ΦBN ` ON`1prq.
§ D formal Z8 ex. sympl., B8 P Rrrr1, . . . , rdss
Z8 ˝ f ˝ Z ´1
8 “ ΦB8.
Birkhoff Normal Forms 7 / 33
The Birkhoff Normal Form Theorem : Formal Integrability. Let f : pR2d, 0q ý or pTd ˆ Rd, T0q ý be a smooth sympl.
§ @N ě 1, D ZN (exact) sympl., BN P RNrr1, . . . , rds
ZN ˝ f ˝ Z ´1
N
“ ΦBN ` ON`1prq.
§ D formal Z8 ex. sympl., B8 P Rrrr1, . . . , rdss
Z8 ˝ f ˝ Z ´1
8 “ ΦB8. § B8 “ BNFpf q : Birkhoff Normal Form (BNF).
Birkhoff Normal Forms 7 / 33
The Birkhoff Normal Form Theorem : Formal Integrability. Let f : pR2d, 0q ý or pTd ˆ Rd, T0q ý be a smooth sympl.
§ @N ě 1, D ZN (exact) sympl., BN P RNrr1, . . . , rds
ZN ˝ f ˝ Z ´1
N
“ ΦBN ` ON`1prq.
§ D formal Z8 ex. sympl., B8 P Rrrr1, . . . , rdss
Z8 ˝ f ˝ Z ´1
8 “ ΦB8. § B8 “ BNFpf q : Birkhoff Normal Form (BNF). § It is unique.
Birkhoff Normal Forms 7 / 33
The Birkhoff Normal Form Theorem : Formal Integrability. Let f : pR2d, 0q ý or pTd ˆ Rd, T0q ý be a smooth sympl.
§ @N ě 1, D ZN (exact) sympl., BN P RNrr1, . . . , rds
ZN ˝ f ˝ Z ´1
N
“ ΦBN ` ON`1prq.
§ D formal Z8 ex. sympl., B8 P Rrrr1, . . . , rdss
Z8 ˝ f ˝ Z ´1
8 “ ΦB8. § B8 “ BNFpf q : Birkhoff Normal Form (BNF). § It is unique. Also Invariant by smooth conjugations.
Birkhoff Normal Forms 7 / 33
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.
Birkhoff Normal Forms 8 / 33
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 : pTd ˆ Rd, T0q ý,
Birkhoff Normal Forms 8 / 33
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 : pTd ˆ Rd, T0q ý,
§ A (lagrangian) invariant torus of f : f -invariant set of the
form ΓS ΓS :“ tpθ, ∇Spθq, θ P Tdu, S : Td Ñ R
Birkhoff Normal Forms 8 / 33
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 : pTd ˆ Rd, T0q ý,
§ A (lagrangian) invariant torus of f : f -invariant set of the
form ΓS ΓS :“ tpθ, ∇Spθq, θ P Tdu, S : Td Ñ R
§ It is a KAM torus if the dynamics of f on ΓS is that of a
(diophantine) translation.
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Birkhoff Normal Forms 9 / 33
Assume that O “ t0u or Td ˆ t0u is a non resonant equilibrium of the smooth symplectic diffeom f : pR2d, 0q ý or pTd ˆ Rd, T0q ý. If B8 “ BNFpf q is non-planar, then the
(KAM) tori.
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Assume that O “ t0u or Td ˆ t0u is a non resonant equilibrium of the smooth symplectic diffeom f : pR2d, 0q ý or pTd ˆ Rd, T0q ý. If B8 “ BNFpf q is non-planar, then the
(KAM) tori. B8 is non-planar or non-degenerate : if Eγ P Rd s.t. @ r, x∇B8prq, γy “ 0.
Birkhoff Normal Forms 9 / 33
Assume that O “ t0u or Td ˆ t0u is a non resonant equilibrium of the smooth symplectic diffeom f : pR2d, 0q ý or pTd ˆ Rd, T0q ý. If B8 “ BNFpf q is non-planar, then the
(KAM) tori. B8 is non-planar or non-degenerate : if Eγ P Rd s.t. @ r, x∇B8prq, γy “ 0.
Assume f is real-analytic and ω0 is Diophantine. If ΦB8 “ Df p0q, then f is integrable : it is real-analytically conjugated to Df p0q in a neighborhood of the origin.
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Birkhoff Normal Forms 10 / 33
‚ Poincar´ e discovered that non integrability is generic : Even if f real analytic the conjugation relation Z8 ˝ f ˝ Z ´1
8 “ ΦBNFpf q
cannot hold with both Z8 and BNFpf q converging.
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‚ Poincar´ e discovered that non integrability is generic : Even if f real analytic the conjugation relation Z8 ˝ f ˝ Z ´1
8 “ ΦBNFpf q
cannot hold with both Z8 and BNFpf q converging. ‚ Siegel (1954) : the formal conjugacy Z8 is in generically divergent.
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‚ Poincar´ e discovered that non integrability is generic : Even if f real analytic the conjugation relation Z8 ˝ f ˝ Z ´1
8 “ ΦBNFpf q
cannot hold with both Z8 and BNFpf q converging. ‚ Siegel (1954) : the formal conjugacy Z8 is in generically divergent. Eliasson’s Question. Are there examples of divergent BNF if f is analytic ?
Birkhoff Normal Forms 10 / 33
‚ Poincar´ e discovered that non integrability is generic : Even if f real analytic the conjugation relation Z8 ˝ f ˝ Z ´1
8 “ ΦBNFpf q
cannot hold with both Z8 and BNFpf q converging. ‚ Siegel (1954) : the formal conjugacy Z8 is in generically divergent. Eliasson’s Question. Are there examples of divergent BNF if f is analytic ?
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§ P´
erez-Marco’s Dichotomy (2003) : @ ω non-resonant fixed, either the BNF is always convergent or it prevalently diverges (Elliptic fixed point, hamiltonian case).
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§ P´
erez-Marco’s Dichotomy (2003) : @ ω non-resonant 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.).
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§ P´
erez-Marco’s Dichotomy (2003) : @ ω non-resonant 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.).
Let d ě 1 and ω0 P Rd non resonant. A generic (prevalent) real analytic symplectic diffeomorphism f : pR2d, 0q ý, defined
divergent BNF. Same result in (AA) case if ω0 is diophantine.
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§ P´
erez-Marco’s Dichotomy (2003) : @ ω non-resonant 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.).
Let d ě 1 and ω0 P Rd non resonant. A generic (prevalent) real analytic symplectic diffeomorphism f : pR2d, 0q ý, defined
divergent BNF. Same result in (AA) case if ω0 is diophantine.
§ By Perez-Marco’s theorem : d “ 1 is enough.
Birkhoff Normal Forms 11 / 33
§ P´
erez-Marco’s Dichotomy (2003) : @ ω non-resonant 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.).
Let d ě 1 and ω0 P Rd non resonant. A generic (prevalent) real analytic symplectic diffeomorphism f : pR2d, 0q ý, defined
divergent BNF. Same result in (AA) case if ω0 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).
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ussmann’s Theorem] If d “ 1 and ω is diophantine is it true that the convergence of the BNF implies that f is integrable in a neighborhood of 0 ?
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ussmann’s Theorem] If d “ 1 and 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
ussmann’s Theorem] If d “ 1 and 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.
topology) by symplectic diffeomorphisms with convergent BNF ?
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Birkhoff Normal Forms 13 / 33
In our appraoch, the divergence of the BNF comes from the following principle :
Birkhoff Normal Forms 13 / 33
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.
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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 : pR2, 0q ý or pT ˆ R, T0q ý has a converging BNF, then it must have much more invariant tori than what a generic
Birkhoff Normal Forms 13 / 33
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 : pR2, 0q ý or pT ˆ R, T0q ý has a converging BNF, then it must have much more invariant tori than what a generic
We illustrate this principle in the case where ω0 is Diophantine with exponent τ : τpωq “ lim sup
kÑ8
´ ln minlPZ |kω ´ l| ln k ă 8.
Birkhoff Normal Forms 13 / 33
For t ą 0 we define
§ Lf ptq : the set of points in tr ă tu which are contained in
an invariant circle Ă tr ă 2tu (r “ p1{2qpx2 ` y 2q)
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For t ą 0 we define
§ Lf ptq : the set of points in tr ă tu which are contained in
an invariant circle Ă tr ă 2tu (r “ p1{2qpx2 ` y 2q)
§ mf ptq “ Leb pTˆs ´ t, trzLf ptqq.
Birkhoff Normal Forms 14 / 33
For t ą 0 we define
§ Lf ptq : the set of points in tr ă tu which are contained in
an invariant circle Ă tr ă 2tu (r “ p1{2qpx2 ` y 2q)
§ mf ptq “ Leb pTˆs ´ t, trzLf ptqq.
Let ω0 be Diophantine. Assume that BNFpf qprq is non-degenerate (B2
r BNFpf qp0q ą 0). Then, if BNFpf qprq
converges mf ptq À exp ˆ ´ ˆ1 t ˙2βpω0q´˙ and βpω0q “
1 1`τpω0q.
Birkhoff Normal Forms 14 / 33
On the other hand
Let ω0 be Diophantine. For a “generic” (prevalent) real analytic symplectic diffeomorphism f : pR2, 0q ý or pT ˆ R, T0q ý with frequency ω0 at 0 and non-degenerate BNF, there exists a sequence tj, lim tj “ 0 such that mf ptjq Á exp ˆ ´ ˆ 1 tj ˙βpω0q`˙ .
Birkhoff Normal Forms 15 / 33
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 pT ˆ R, T0q ý f pθ, rq “ pθ ` 2πω, rq ` pOprq, Opr 2qq, ω Diophantine “ ΦΩ ˝ fF, Ωprq “ 2πω0r ` b2r 2, Fpθ, rq “ Opr 3q
Sketch of the proof of Theorem 1 17 / 33
We assume pT ˆ R, T0q ý f pθ, rq “ pθ ` 2πω, rq ` pOprq, Opr 2qq, ω Diophantine “ ΦΩ ˝ fF, Ωprq “ 2πω0r ` b2r 2, Fpθ, rq “ Opr 3q b2 ‰ 0, Ω P OσpDp0, ¯ ρqq, F P OσpTh ˆ Dp0, ¯ ρqq : real-symmetric (wrt compl. conj.) holomorphic.
Sketch of the proof of Theorem 1 17 / 33
We assume pT ˆ R, T0q ý f pθ, rq “ pθ ` 2πω, rq ` pOprq, Opr 2qq, ω Diophantine “ ΦΩ ˝ fF, Ωprq “ 2πω0r ` b2r 2, Fpθ, rq “ Opr 3q b2 ‰ 0, Ω P OσpDp0, ¯ ρqq, F P OσpTh ˆ Dp0, ¯ ρqq : real-symmetric (wrt compl. conj.) holomorphic. We can assume for some a ą 0 (apply BNF up to some order) sup
|ℑθ|ăh,rPDp0,¯ ρq
|Fpθ, rq| :“ }F}h,¯
ρ ď ¯
ρ a. Fix 0 ă ρ ă ¯ ρ.
Sketch of the proof of Theorem 1 17 / 33
Sketch of the proof of Theorem 1 18 / 33
We define various approximate Normal Forms : pg ˚q´1 ˝ ΦΩprq ˝ fF ˝ g ˚ “ ΦΩ˚prq ˝ fF ˚, F ˚ ! 1 }F ˚} À expp´p1{ρq2βpω0q´q defined on various domains of the form Wh,U “ Th ˆ U˚, U˚
Sketch of the proof of Theorem 1 18 / 33
We define various approximate Normal Forms : pg ˚q´1 ˝ ΦΩprq ˝ fF ˝ g ˚ “ ΦΩ˚prq ˝ fF ˚, F ˚ ! 1 }F ˚} À expp´p1{ρq2βpω0q´q defined on various domains of the form Wh,U “ Th ˆ U˚, U˚
§ r˚ “ BNFs : Approximate Birkhoff Normal Forms defined
in Th ˆ Dp0, ρbq (b “ τ ` 2).
Sketch of the proof of Theorem 1 18 / 33
We define various approximate Normal Forms : pg ˚q´1 ˝ ΦΩprq ˝ fF ˝ g ˚ “ ΦΩ˚prq ˝ fF ˚, F ˚ ! 1 }F ˚} À expp´p1{ρq2βpω0q´q defined on various domains of the form Wh,U “ Th ˆ U˚, U˚
§ r˚ “ BNFs : Approximate Birkhoff Normal Forms defined
in Th ˆ Dp0, ρbq (b “ τ ` 2).
§ r˚ “ KAMs : Approximate KAM Normal Forms defined
holes.
Sketch of the proof of Theorem 1 18 / 33
We define various approximate Normal Forms : pg ˚q´1 ˝ ΦΩprq ˝ fF ˝ g ˚ “ ΦΩ˚prq ˝ fF ˚, F ˚ ! 1 }F ˚} À expp´p1{ρq2βpω0q´q defined on various domains of the form Wh,U “ Th ˆ U˚, U˚
§ r˚ “ BNFs : Approximate Birkhoff Normal Forms defined
in Th ˆ Dp0, ρbq (b “ τ ` 2).
§ r˚ “ KAMs : Approximate KAM Normal Forms defined
holes.
§ r˚ “ HJs : For each hole D of UKAM we find disks
ˆ D Ą D, ˇ D Ă ˆ D and an approximate Hamilton-Jacobi Normal Form in Th ˆ pˆ Dzˇ Dq.
Sketch of the proof of Theorem 1 18 / 33
We define various approximate Normal Forms : pg ˚q´1 ˝ ΦΩprq ˝ fF ˝ g ˚ “ ΦΩ˚prq ˝ fF ˚, F ˚ ! 1 }F ˚} À expp´p1{ρq2βpω0q´q defined on various domains of the form Wh,U “ Th ˆ U˚, U˚
§ r˚ “ BNFs : Approximate Birkhoff Normal Forms defined
in Th ˆ Dp0, ρbq (b “ τ ` 2).
§ r˚ “ KAMs : Approximate KAM Normal Forms defined
holes.
§ r˚ “ HJs : For each hole D of UKAM we find disks
ˆ D Ą D, ˇ D Ă ˆ D and an approximate Hamilton-Jacobi Normal Form in Th ˆ pˆ Dzˇ Dq. KAM overlaps with BNF and HJ.
Sketch of the proof of Theorem 1 18 / 33
are constructed by successive conjugations on smaller and smaller complex domains Thi ˆ Ui fYi ˝ ΦΩi ˝ fFi ˝ f ´1
Yi
“ ΦΩi`1 ˝ fFi`1 Fi`1 “ O2pFiq Ωi`1prq “ Ωiprq ` p2πq´1 ż
T
Fipθ, rqdθ.
Sketch of the proof of Theorem 1 19 / 33
are constructed by successive conjugations on smaller and smaller complex domains Thi ˆ Ui fYi ˝ ΦΩi ˝ fFi ˝ f ´1
Yi
“ ΦΩi`1 ˝ fFi`1 Fi`1 “ O2pFiq Ωi`1prq “ Ωiprq ` p2πq´1 ż
T
Fipθ, rqdθ. by solving the (truncated) linearized equation Fi ´ p2πq´1 ż
T
Fipθ, ¨qdθ “ Yi ˝ ΦΩi ´ Yi (4)
Sketch of the proof of Theorem 1 19 / 33
are constructed by successive conjugations on smaller and smaller complex domains Thi ˆ Ui fYi ˝ ΦΩi ˝ fFi ˝ f ´1
Yi
“ ΦΩi`1 ˝ fFi`1 Fi`1 “ O2pFiq Ωi`1prq “ Ωiprq ` p2πq´1 ż
T
Fipθ, rqdθ. by solving the (truncated) linearized equation Fi ´ p2πq´1 ż
T
Fipθ, ¨qdθ “ Yi ˝ ΦΩi ´ Yi (4)
Sketch of the proof of Theorem 1 19 / 33
are constructed by successive conjugations on smaller and smaller complex domains Thi ˆ Ui fYi ˝ ΦΩi ˝ fFi ˝ f ´1
Yi
“ ΦΩi`1 ˝ fFi`1 Fi`1 “ O2pFiq Ωi`1prq “ Ωiprq ` p2πq´1 ż
T
Fipθ, rqdθ. by solving the (truncated) linearized equation Fi ´ p2πq´1 ż
T
Fipθ, ¨qdθ “ Yi ˝ ΦΩi ´ Yi (4)
§ For KAM : avoid resonances when solving (4)
|k∇Ωiprq ´ 2πl| ě K ´1
i
, @ 0 ă |k| ă Ni (5) hence Ui`1 “ Uiz Ť disks, N2
i disks radii K ´1 i
Sketch of the proof of Theorem 1 19 / 33
are constructed by successive conjugations on smaller and smaller complex domains Thi ˆ Ui fYi ˝ ΦΩi ˝ fFi ˝ f ´1
Yi
“ ΦΩi`1 ˝ fFi`1 Fi`1 “ O2pFiq Ωi`1prq “ Ωiprq ` p2πq´1 ż
T
Fipθ, rqdθ. by solving the (truncated) linearized equation Fi ´ p2πq´1 ż
T
Fipθ, ¨qdθ “ Yi ˝ ΦΩi ´ Yi (4)
§ For KAM : avoid resonances when solving (4)
|k∇Ωiprq ´ 2πl| ě K ´1
i
, @ 0 ă |k| ă Ni (5) hence Ui`1 “ Uiz Ť disks, N2
i disks radii K ´1 i § For (approx.) BNF Ui are smaller and smaller disks
centered at 0 (essentially no resonances).
Sketch of the proof of Theorem 1 19 / 33
b
ΩKAM Ξ “ BNF D hole of KAM Dp0, ρq Dp0, ρbq ΩBNF on R-axis
Figure: KAM and BNF Normal Forms
Sketch of the proof of Theorem 1 20 / 33
If one faces a resonance at the i-th step of KAM : (5) fails p2πq´1BΩpcq “ l{k, |k| ă Ni.
Sketch of the proof of Theorem 1 21 / 33
If one faces a resonance at the i-th step of KAM : (5) fails p2πq´1BΩpcq “ l{k, |k| ă Ni.
§ Make a resonant NF (similar to BNF) to eliminate
harmonics R kZ : can be done on Th{3 ˆ Dpc, ˆ K ´1
i
q ˆ K ´1
i
“ N´ ln Ni
i
. (Compare to K ´1
i
“ expp´Ni{pln Niqaq).
Sketch of the proof of Theorem 1 21 / 33
If one faces a resonance at the i-th step of KAM : (5) fails p2πq´1BΩpcq “ l{k, |k| ă Ni.
§ Make a resonant NF (similar to BNF) to eliminate
harmonics R kZ : can be done on Th{3 ˆ Dpc, ˆ K ´1
i
q ˆ K ´1
i
“ N´ ln Ni
i
. (Compare to K ´1
i
“ expp´Ni{pln Niqaq).
§ Rescale (covering) and get a system very close to a
hamiltonian in Tkh{3 ˆ Dp0, k ˆ K ´1
i
q : Pendulum like.
Sketch of the proof of Theorem 1 21 / 33
If one faces a resonance at the i-th step of KAM : (5) fails p2πq´1BΩpcq “ l{k, |k| ă Ni.
§ Make a resonant NF (similar to BNF) to eliminate
harmonics R kZ : can be done on Th{3 ˆ Dpc, ˆ K ´1
i
q ˆ K ´1
i
“ N´ ln Ni
i
. (Compare to K ´1
i
“ expp´Ni{pln Niqaq).
§ Rescale (covering) and get a system very close to a
hamiltonian in Tkh{3 ˆ Dp0, k ˆ K ´1
i
q : Pendulum like.
§ 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
If one faces a resonance at the i-th step of KAM : (5) fails p2πq´1BΩpcq “ l{k, |k| ă Ni.
§ Make a resonant NF (similar to BNF) to eliminate
harmonics R kZ : can be done on Th{3 ˆ Dpc, ˆ K ´1
i
q ˆ K ´1
i
“ N´ ln Ni
i
. (Compare to K ´1
i
“ expp´Ni{pln Niqaq).
§ Rescale (covering) and get a system very close to a
hamiltonian in Tkh{3 ˆ Dp0, k ˆ K ´1
i
q : Pendulum like.
§ 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
One gets a NF defined on Th{20 ˆ ˆ Dzˇ D ˆ D “ Dpc, ˆ K ´1
i
q, ˇ D corresponds to the eye
Sketch of the proof of Theorem 1 22 / 33
One gets a NF defined on Th{20 ˆ ˆ Dzˇ D ˆ D “ Dpc, ˆ K ´1
i
q, ˇ D corresponds to the eye g ´1
i
˝ ΦΩi ˝ fFi ˝ gi “ ΦΩHJ
D ˝ fF HJ D . Sketch of the proof of Theorem 1 22 / 33
b
ΩKAM Ξ “ BNF D hole of KAM Dp0, ρq Dp0, ρbq ΩBNF on R-axis
Figure: KAM and BNF Normal Forms
Sketch of the proof of Theorem 1 23 / 33
b
ΩHJ
ˆ Dz ˇ D on ˆ
Dzˇ D ΩKAM Ξ “ BNF D hole of KAM Dp0, ρq Dp0, ρbq ΩBNF on R-axis
Figure: The various Normal Forms
Sketch of the proof of Theorem 1 24 / 33
A priori one might think there is no gain in doing this.
Sketch of the proof of Theorem 1 25 / 33
A priori one might think there is no gain in doing this. Interest :
Sketch of the proof of Theorem 1 25 / 33
A priori one might think there is no gain in doing this. Interest : One knows how this Hamilton-Jacobi NF is
Sketch of the proof of Theorem 1 25 / 33
A priori one might think there is no gain in doing this. Interest : One knows how this Hamilton-Jacobi NF is
Extension Principle If there exists a holomorphic function Ξ P Opˆ Dq such that }ΩHJ
D ´ Ξ}p4{5qˆ Dzp1{5qˆ D À ν
then radiuspˇ Dq À ν1{43.
Sketch of the proof of Theorem 1 25 / 33
A priori one might think there is no gain in doing this. Interest : One knows how this Hamilton-Jacobi NF is
Extension Principle If there exists a holomorphic function Ξ P Opˆ Dq such that }ΩHJ
D ´ Ξ}p4{5qˆ Dzp1{5qˆ D À ν
then radiuspˇ Dq À ν1{43. Amounts to Residue Principle applied to pz2 ` a2q1{2.
Sketch of the proof of Theorem 1 25 / 33
These Normal Forms almost coincide on their domain of definitions :
Sketch of the proof of Theorem 1 26 / 33
These Normal Forms almost coincide on their domain of definitions : Matching Principle If on an annulus like domain g ´1
j
˝ ΦΩprq ˝ fF ˝ gj “ ΦΩprqi ˝ fFj, j “ 1, 2 where Fj are small then ∇Ω1 and ∇Ω2 coincide with good approximation.
Sketch of the proof of Theorem 1 26 / 33
These Normal Forms almost coincide on their domain of definitions : Matching Principle If on an annulus like domain g ´1
j
˝ ΦΩprq ˝ fF ˝ gj “ ΦΩprqi ˝ fFj, j “ 1, 2 where Fj are small then ∇Ω1 and ∇Ω2 coincide with good approximation.
§ ΩBNF « ΩKAM on Dp0, ρbq.
Sketch of the proof of Theorem 1 26 / 33
These Normal Forms almost coincide on their domain of definitions : Matching Principle If on an annulus like domain g ´1
j
˝ ΦΩprq ˝ fF ˝ gj “ ΦΩprqi ˝ fFj, j “ 1, 2 where Fj are small then ∇Ω1 and ∇Ω2 coincide with good approximation.
§ ΩBNF « ΩKAM on Dp0, ρbq. § ΩKAM « ΩHJ ˆ Dz ˇ D on ˆ
Dzˇ D
Sketch of the proof of Theorem 1 26 / 33
These Normal Forms almost coincide on their domain of definitions : Matching Principle If on an annulus like domain g ´1
j
˝ ΦΩprq ˝ fF ˝ gj “ ΦΩprqi ˝ fFj, j “ 1, 2 where Fj are small then ∇Ω1 and ∇Ω2 coincide with good approximation.
§ ΩBNF « ΩKAM on Dp0, ρbq. § ΩKAM « ΩHJ ˆ Dz ˇ D on ˆ
Dzˇ D Furthermore if the BNF converges on Dp0, 1q : ΩBNF « Ξ “ BNF, on Dp0, ρbq.
Sketch of the proof of Theorem 1 26 / 33
Sketch of the proof of Theorem 1 27 / 33
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
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
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 ΩKAM « Ξ “ BNF
Sketch of the proof of Theorem 1 27 / 33
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 ΩKAM « Ξ “ BNF
and thus from the Matching Property ΩHJ
ˆ Dz ˇ D « Ξ “ BNF
Dzˇ D.
Sketch of the proof of Theorem 1 27 / 33
Precise form :
Sketch of the proof of Theorem 1 28 / 33
Precise form : If
§ U “ Dp0, ρqzpŤ 1ďjďN Dpzj, ǫjqq § Dp0, σq Ă U § f P OpUq satisfies }f }U ď 1,
}f }BB ď m. Then @ z P ˆ U :“ Dp0, ρqzpŤ
1ďjďN Dpzj, djqq, 2ǫj ă dj ă 1
ln |f pzq| ď ˆlnp|z|{ρq lnpσ{ρq ´
N
ÿ
j“1
lnpdj{2ρq lnpǫj{ρq ˙ ln m (6)
Sketch of the proof of Theorem 1 28 / 33
Precise form : If
§ U “ Dp0, ρqzpŤ 1ďjďN Dpzj, ǫjqq § Dp0, σq Ă U § f P OpUq satisfies }f }U ď 1,
}f }BB ď m. Then @ z P ˆ U :“ Dp0, ρqzpŤ
1ďjďN Dpzj, djqq, 2ǫj ă dj ă 1
ln |f pzq| ď ˆlnp|z|{ρq lnpσ{ρq ´
N
ÿ
j“1
lnpdj{2ρq lnpǫj{ρq ˙ ln m (6) Typically fails if one has N2 holes of size e´N (OK if N1{2 holes).
Sketch of the proof of Theorem 1 28 / 33
Precise form : If
§ U “ Dp0, ρqzpŤ 1ďjďN Dpzj, ǫjqq § Dp0, σq Ă U § f P OpUq satisfies }f }U ď 1,
}f }BB ď m. Then @ z P ˆ U :“ Dp0, ρqzpŤ
1ďjďN Dpzj, djqq, 2ǫj ă dj ă 1
ln |f pzq| ď ˆlnp|z|{ρq lnpσ{ρq ´
N
ÿ
j“1
lnpdj{2ρq lnpǫj{ρq ˙ ln m (6) Typically fails if one has N2 holes of size e´N (OK if N1{2 holes). Ý Ñ Cannot push the KAM scheme too far :
Sketch of the proof of Theorem 1 28 / 33
Precise form : If
§ U “ Dp0, ρqzpŤ 1ďjďN Dpzj, ǫjqq § Dp0, σq Ă U § f P OpUq satisfies }f }U ď 1,
}f }BB ď m. Then @ z P ˆ U :“ Dp0, ρqzpŤ
1ďjďN Dpzj, djqq, 2ǫj ă dj ă 1
ln |f pzq| ď ˆlnp|z|{ρq lnpσ{ρq ´
N
ÿ
j“1
lnpdj{2ρq lnpǫj{ρq ˙ ln m (6) Typically fails if one has N2 holes of size e´N (OK if N1{2 holes). Ý Ñ Cannot push the KAM scheme too far : Adapted Normal Form.
Sketch of the proof of Theorem 1 28 / 33
b
ΩHJ
ˆ Dz ˇ D on ˆ
Dzˇ D ΩKAM Ξ “ BNF D hole of KAM Dp0, ρq Dp0, ρbq ΩBNF on R-axis
Figure: Adapted Normal Forms
Sketch of the proof of Theorem 1 29 / 33
Since
§ ΩKAM « ΩBNF « BNF “ holomorphic (Matching
Principle)
§ ΩKAM « ΩHJ D (No-Screening Principle) § ΩHJ D satisify the Extension Principle
all the holes ˇ D of the HJ Normal Form have to be “very small”.
Sketch of the proof of Theorem 1 30 / 33
Use the classical KAM estimates on the measure of the set of invariant curves on UKAM and all the ˆ Dzˇ D : mΦΩ˝fFpp3{4qρq À expp´p1{ρq2βpωq´q ` ÿ
DPDρ
|pˇ D X Rq| À expp´p1{ρq2βpωq´q.
Sketch of the proof of Theorem 1 31 / 33
https://arxiv.org/pdf/1906.01096.pdf
Sketch of the proof of Theorem 1 32 / 33
Sketch of the proof of Theorem 1 33 / 33