Symplectic Homogenization

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Symplectic Homogenization

Claude Viterbo

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Claude Viterbo. Symplectic Homogenization. 2007. �hal-00201500�

hal-00201500, version 1 - 30 Dec 2007

Symplectic Homogenization

Claude Viterbo

Centre de Math´ematiques Laurent Schwartz.

UMR 7640 du CNRS

Ecole Polytechnique - 91128 Palaiseau, France ´ [email protected] ^∗

31st December 2007

Abstract

Let H(q, p) be a Hamiltonian on T

^∗

T

ⁿ

. We show that the se- quence H

_k

(q, p) = H(kq, p) converges for the γ topology defined by the author, to H(p). This is extended to the case where only some of the variables are homogenized, that is the sequence H(kx, y, q, p) where the limit is of the type H(y, q, p) and thus yields an “effective Hamiltonian”. We give here the proof of the convergence, and the first properties of the homogenization operator, and give some immediate consequences for solutions of Hamilton-Jacobi equations, construction of quasi-states, etc....

We also prove that the function H coincides with Mather’s α function defined in [Ma] which gives a new proof of its symplectic invariance proved by P. Bernard [Bern 2].

Keywords: Homogenization, symplectic topology, Hamiltonian flow, Hamilton-Jacobi equation, variational solutions.

AMS Classification: primary: 37J05, 53D35 secondary: 35F20, 49L25, 37J40, 37J50.

∗

Supported also by Agence Nationale de la Recherche projects Symplexe and GRST.

Contents

1 Introduction 3

2 A crash course on generating function metric 5

2.1 Variational solutions of Hamilton-Jacobi equations . . . . 9

3 Statement of the main results 9 3.1 Standard homogenization. . . . 9

3.2 Partial Homogenization . . . 11

3.3 Homogenized Hamilton-Jacobi equations . . . 12

4 Proof of the main theorem. 13 4.1 Reformulating the problem and finding the homogenized Hamil- tonian. . . 14

4.2 Concluding the proof of the main theorem. . . 21

5 Proof of theorem 3.2. 30 6 Proof of the partial homogenization case 32 6.1 Resolution in the (q

2

, P

1

) variables . . . 35

6.2 Generating functions for H(kx, y, q, p) . . . 36

6.3 The case of ℓ terms . . . 36

7 Proof of proposition 3.6 39 8 Non compact-supported Hamiltonians and the time depen- dent case 40 8.1 The coercive case . . . 40

8.2 The non coercive case . . . 42

9 Homogenization in the p variable 43 9.1 Partial Legendre transform . . . 43

9.2 Connection with Mather α function . . . 46

10 Some examples and applications 47 10.1 Homogenization of H(t, q, p) in the variable t . . . 47

10.2 The one dimensional case:

¹₂

| p |

²

− V (x) . . . 48

10.3 Some computations . . . 49

10.4 Homogenized metric and the Thurston-Gromov norm . . . 50

11 Further questions 51

Appendix: Capacity of completely integrable systems. 52

References 53

1 Introduction

The aim of this paper is to define the notion of homogenization for a Hamil- tonian diffeomorphism of T

^∗

T

ⁿ

. In other words, given H(t, q, p) defined for (q, p) in T

^∗

T

ⁿ

, t in R , 1-periodic in t we shall study whether the sequence H

_k

defined by H

k

(t, q, p) = H(kt, kq, p) converges for the symplectic metric γ de- fined in [V1] to some Hamiltonian H, necessarily of the form H(q, p) = h(p).

This convergence of H

_k

to H should be understood as the convergence of the time one flows of H

k

, ϕ

^t_k

to the time one flow of H, ϕ

^t

-again for the symplectic metric γ. This metric is necessarily rather weak, since for example there cannot be any C

⁰

convergence for the flows.

However such convergence implies convergence of the variational solution (see [O-V] for the definition) of Hamilton-Jacobi equations.

(HJ)

_∂

∂t

u(t, q) + H(k · t, k · q,

_∂x^∂

u(t, x)) = 0 u(0, q) = f (q) .

to the variational solutions of

(HJ)

_∂

∂t

u(t, q) + H(t, q,

_∂x^∂

u(t, x)) = 0 u(0, q) = f(q) .

It is important to notice that none of these convergences implies any kind of pointwise or almost everywhere convergence (however C

⁰

convergence of the flows implies γ-convergence as we proved in [V1], we refer to Humili`ere’s work in [Hu] for stronger statements), but rather to some variational notions of convergence, like Γ-convergence (see [de G], [Dal M]). This or similar notions are used in homogenization theory, the theory of viscosity solutions for Hamilton-Jacobi equations (see [L-P-V]), or the rescaling of metrics on T

ⁿ

(see [Gr]).

All the above-mentioned papers can be considered as anticipating the theory of “symplectic homogenization” that is presented here. We believe some of the advantages of this unified treatment are

(1) the disposal of any convexity or even coercivity (as in [L-P-V]) assump-

tion on H in the p direction, usually needed to define H because of the

use of minimization techniques for the Lagrangian. In fact our homog- enization is defined on compact supported objects, and then showed to extend to a number of non compact supported situations.

(2) the natural extension of homogenization to cases where H has very little regularity (less than continuity is needed).

(3) a well defined and common definition of the convergence of H

k

to H or ϕ

k

to ϕ that applies to flows, Hamilton-Jacobi equations, etc.

(4) The symplectic invariance of the homogenized Hamiltonian extends the invariance results defined for example in [Bern 1] for Mather’s α function, making his constructions slightly less mysterious.

(5) geometric properties of the function H (see proposition 3.2, (5)). yield- ing computational methods extending those obtained in the one-dimensional case in [L-P-V] or in other cases (see for example [Conc]).

This paper will address these fundamental questions, some other appli- cations will be dealt with in subsequent papers.

Acknowledgements

This paper originated in the attempt to understand the physical phenomenon of self-adaptative oscillators described in [B-C-B], that the author learned from a popularization CNRS journal “Images de la physique”. We hope to be able to explain in a future publication the connection between Symplectic homogenization and self-adaptative phenomena in vibrating structures. We not only wish to thank the authors of [B-C-B] for writing the paper and its popularized version[Bou], but also CNRS for maintaining journals like

“Images de la physique”.

I wish to thank S. Kuksin and P. Pansu for useful discussions and the

University of Beijing for hospitality during the preparation of this work. I

am also grateful to the organizers of the Yashafest in Stanford, and the

Conference in symplectic topology in Kyoto for giving me the opportunity

to present preliminary versions of this work. I also warmly thank Claudine

Harmide for her efficient and speedy typing.

2 A crash course on generating function met- ric

This section is devoted to defining the metric γ for which we shall later prove that the sequence H

k

(q, p) = H(kq, p) is convergent. The reader may well skip this section and jump directly to section 3, possibly returning here for reference.

Let M be an n-dimensional closed manifold, L be a Lagrangian subman- ifold in T

^∗

M Hamiltonianly isotopic to the zero section O

_M

(i.e. there is a Hamiltonian isotopy ϕ

t

such that ϕ

1

(O

M

) = L).

Definition 2.1. The smooth function S : M × R

^k

→ R is a generating function quadratic at infinity for L if

i) there is a non degenerate quadratic form q on R

^q

such that

|∇

^ξ

S(q; ξ) − ∇ B(ξ) | ≤ C ii) the map

(q; ξ) → ∂S

∂ξ (q; ξ) has zero as a regular value

iii) by i) and ii), Σ

_s

= { (q; ξ) |

^∂S_∂ξ

(q; ξ) = 0 } is a compact submanifold in M × R

^k

. The map

i

δ

: Σ

s

→ T

^∗

M (q; ξ) → (q, ∂S

∂q (q; ξ)) has image i

s

(Σ

s

) = L.

Remarks 2.2. (1) In this paper, we shall always use a semicolon to separate the “base variables” q from the “fibre variables”, ξ.

(2) We still speak of generating function when there are no fibre variables.

In this case, L is the graph of the differential dS(q).

(3) In the sequel we shall abbreviate “generating function quadratic at infinity” by “G.F.Q.I.”.

Now when L is Hamiltonianly isotopic to the zero section, we know ac- cording to [V1] (prop 1.5, page 688) that the generating function is unique up to some elementary operations (see [V1], loc.cit.). Moreover, an elementary computation shows that denoting by S

^λ

the set

{ (q; ξ) ∈ M × R

^k

| S(q; ξ) ≤ λ }

we have for C large enough that

H

^∗

(S

^c

, S

^−c

) ≃ H

^∗

(M ) ⊗ H

^∗

(D

⁻

, ∂D

⁻

)

where D

⁻

is the unit disc of the negative eigenspace of B. Therefore, to each cohomology class α in H

^∗

(M) − { 0 } we may associate the image of α ⊗ T (T is a chosen generator of H

^∗

(D

⁻

, ∂D

⁻

) ≃ Z ), and by minmax a critical level c(α, S) (see [V1] section 2, p.690-693).

Definition 2.3. Let L be Hamiltonianly isotopic to the zero section. We set c

−

(L) = c(1, S) 1 ∈ H

⁰

(M)

c

₊

(L) = c(µ, S) µ ∈ H

ⁿ

(M ) \ { 0 } γ(L) = c(µ, S ) − c(1, S)

Remark 2.4. 1) Note that we may always add a constant to S. This shifts c

−

(L) and c

+

(L) by the same constant, so that, unless we normalized in some way S, c

₋

(L), c

₊

(L) are not well defined, but their difference γ(L) is well-defined. However, if we specify the Hamiltonian H yielding the isotopy between the zero section and L, we may normalize S by requiring that its critical values coincide with the actions

Z

0¹

[p(t) ˙ q(t) − H(t, q(t), p(t))] dt where (q(t), p(t)) = ϕ

^t

(q(0), 0) satisfies p(1) = 0.

Thus c

±

(H) is well defined. Since if ϕ

¹

is generated by some compact supported Hamiltonian, such a Hamiltonian is unique, we may define c

±

(ϕ) for ϕ ∈ H D

c

(T

^∗

T

ⁿ

)

2) We shall sometimes deal with the case M = R

ⁿ

. Then we need quadraticity of S both in the ξ and x variable, so that

i) in Definition 2.1 should be replaced by

i’) there exists a quadratic form q(x, ξ) on M × R

^k

(= R

ⁿ

× R

^k

) such that

|∇ S(q; ξ) − ∇ B(q; ξ) | ≤ C .

The map γ is well defined on the set L of Lagrangian submanifolds Hamilto- nianly isotopic to the zero section, where the Hamiltonian is assumed to be compact supported.

According to [V1], the metric γ defines a metric on L by setting

Definition 2.5. γ(L, L

^′

) = c(µ, S ⊖ S

^′

) − c(1, S ⊖ S

^′

) where (S ⊖ S

^′

)(q; ξ, ξ

^′

) =

S(q; ξ) − S(q; ξ

^′

).

That this is indeed a metric is a consequence of Lusternik-Shnirelman’s theory, as we proved in [V1].

Our goal however is to define a metric on H

^c

(T

^∗

M) = C

_c^∞

([0, 1] × T

^∗

T

ⁿ

, R ) the set of compact supported, time dependent Hamiltonian isotopies of T

^∗

M , and on HD

c

(T

^∗

M ) the group of time one maps of Hamiltonians in H

c

(T

^∗

M).

In general, we may set Definition 2.6. We set

b

γ(ϕ) = sup { γ(ϕ(L), L) | L ∈ L} . b

γ(ϕ, ψ) = b γ(ϕψ

⁻¹

).

Remark 2.7. For M = T

ⁿ

, the graph of ϕ ,

Γ(ϕ) = { (z, ϕ(z)) | z ∈ T

^∗

T

ⁿ

}

is a Lagrangian submanifold of T

^∗

T

ⁿ

× T

^∗

T

ⁿ

(where T

^∗

M is T

^∗

M with the symplectic form of opposite sign : − dp ∧ dq).

But T

^∗

T

ⁿ

× T

^∗

T

ⁿ

is covered by T

^∗

(∆

T^∗Tⁿ

) where ∆

T^∗Tⁿ

is the diagonal, and we may lift Γ(ϕ) to e Γ(ϕ), which is now a Lagrangian submanifold in T

^∗

(∆

T^∗Tⁿ

).

When ϕ has compact support, we may compactify both e Γ(ϕ) and ∆

T^∗Tⁿ

and we get a Lagrangian submanifold Γ(ϕ) in T

^∗

(S

ⁿ

× T

ⁿ

). We then defined in [V1] (page 679)

γ(ϕ) = c(µ

Tⁿ

⊗ µ

Sⁿ

, Γ(ϕ)) − c(1 ⊗ 1, Γ(ϕ)) . We proved in [V3] that b γ(ϕ) ≤ γ(ϕ).

Proposition 2.8. (see [V3])

The map b γ defines a bi-invariant metric on HD

^c

(T

^∗

M) since i) it is nondegenerate b γ(ϕ) = 0 ⇐⇒ ϕ = id

ii) it is invariant by conjugation γ(ψϕψ b

⁻¹

) = b γ (ϕ) for any ψ in HD

^c

(T

^∗

M ).

iii) it satisfies the triangle inequality b

γ(ϕψ) ≤ b γ(ϕ) + b γ(ψ) for any ϕ, ψ in HD

^c

(T

^∗

M).

The above properties also hold for γ instead of b γ when M = T

ⁿ

.

Let λ = pdq be the Liouville form on T

^∗

M . A vector field Z is called a Liouville vector field, if Z = Z

_λ

+ X

_H

where X

_H

is Hamiltonian, while i

Zλ

ω = λ (hence i

Z

ω = λ + dH). In particular Z is conformal (i.e. the flow ψ

t

of Z satisfies ψ

_t^∗

ω = e

^t

ω) and we have

(1) γ(ψ

t

ϕψ

_t⁻¹

) = e

^t

γ(ϕ) .

In the set H

^c

(T

^∗

M ) the metric b γ is defined as follows

Definition 2.9. Let H be a Hamiltonian, with flow ψ

^t

. We denote by b

γ(H) = sup b

γ(ψ

^t

) | t ∈ [0, 1]

and similarly

b

γ(H) = sup b

γ(ψ

^t

) | t ∈ [0, 1]

Finally we state some convergence criterion for the b γ metric.

Proposition 2.10. Let M = R

ⁿ

or T

ⁿ

.

1) Assume the sequence H

k

of Hamiltonians on T

^∗

M with fixed support converges C

⁰

to H. Then H

k

converges for the γ metric to H.

2) There is a constant C such that if ϕ is supported in W

R

= { (q, p) ∈ T

^∗

M | | p | ≤ R } and for any z in T

^∗

M d(z, ϕ(z)) ≤ ε, we have

γ(ϕ) ≤ CεR

Proof. Part 1) is proved as in [V1] Proposition 4.6 (page 699).

As for part 2), we may follow the some pattern as in the proof of propo- sition 4.15 (loc.cit. page 703), provided we prove that there is a C

²

small Hamiltonian supported in W

2R

such that its time one map, ψ, satisfies

d(z, ψ(z)) ≥ ε ∀ z ∈ W

R

.

Given H without critical point in W

R

, and C

²

small, it is well known that ψ

^t

has no periodic orbit of period less than 1, hence d(z, ψ(z)) is bounded from below by some ε

0

> 0. This concludes our proof.

We may therefore define, as Humili`ere did in [Hu], the completion c H (T

^∗

M) of H

c

(T

^∗

M ) for γ. By the above proposition we deduce that a C

⁰

-converging sequence of Hamiltonians will be a Cauchy sequence for γ, hence defines an element in c H (T

^∗

M). We thus get

Proposition 2.11. There is an inclusion map

C

_c⁰

([0, 1] × T

^∗

M, R ) → c H (T

^∗

M )

Similarly if HD

0

(T

^∗

M ) is the C

⁰

closure of HD

c

(T

^∗

M) we have an inclusion

HD

⁰

(T

^∗

M) → HD d (T

^∗

M )

2.1 Variational solutions of Hamilton-Jacobi equations

Let us consider the symplectic covering of T

^∗

T

ⁿ

× T

^∗

T

ⁿ

by T

^∗

(T

ⁿ

× R

ⁿ

) given by

(q, p, Q, P ) −→ (Q, p, p − P, Q − q)

Let ϕ

^t

be the Hamiltonian flow of H(q, p). Then, the graph of ϕ

^t

has image e Γ(ϕ

^t

). Let S

t

(q, P, ξ) be a generating function for Γ(ϕ e

^t

). We denote by c(1(q) ⊗ µ, S

t

) the number c(µ, S

t,q

) where S

t,q

(P, ξ ) = S(q, P, ξ). Then u

t

(q, P ) = c(1(q) ⊗ 1(P ), S

t

) is a variational solution of

_∂

∂t

u

t

(q, P ) + H(q, P +

_∂t^∂

u

t

(q)) = 0 u

0

(q, P ) = 0

We refer to [O-V] and [C-V] for more informations on variational solu- tions.

3 Statement of the main results

3.1 Standard homogenization.

Let H(t, q, p) be a C

²

Hamiltonian on T

^∗

T

ⁿ

, 1-periodic in t, and compact supported

Theorem 3.1 (Main theorem).

Let H (t, q, p) be a C

²

Hamiltonian on the torus T

ⁿ

. Then the following holds:

(1) The sequence H

k

(t, q, p) = H(kt, kq, p) γ-converges to H(t, q, p) = h(p), where h is continuous.

(2) The function H only depends on ϕ

¹

, the time one map associated to H (i.e. it does not depends on the isotopy (ϕ

^t

)

t∈[0,1]

).

(3) The map

A : C

²

([0, 1] × T

^∗

T

ⁿ

, R ) → C

⁰

( R

ⁿ

, R )

given by A (H) = H extends to a nonlinear projector (i.e. it satisfies A

²

= A ) with Lipschitz constant 1

A : c H (T

^∗

T

ⁿ

) → C

⁰

( R

ⁿ

, R )

where the metric on c H is given by γ and the metric on C

⁰

( R

ⁿ

, R ) is the C

⁰

metric.

The next theorem states some properties of the map A

Theorem 3.2 (Main properties of symplectic homogenization).

Let A be the map defined in the above theorem. Then it satisfies the following properties:

(1) The map A is monotone, i.e. if H

₁

≤ H

₂

then A (H

₁

) ≤ A (H

₂

).

(2) The map A is invariant by Hamiltonian symplectomorphism:

A (H ◦ ψ) = A (H) for all ψ ∈ HD (T

^∗

T

ⁿ

) (3) We have A ( − H) = −A (H).

(4) The map A extends to characteristic functions of subsets, hence in- duces a map (still denoted by A ) between the set of subsets of T

^∗

T

ⁿ

, P (T

^∗

T

ⁿ

), to the set of subsets of R

ⁿ

, P ( R

ⁿ

). This map is bounded by the symplectic shape of Sikorav (see [Be, Sik, El]), i.e.

shape(U ) ⊂ A (U )

(5) If L is a Lagrangian Hamiltonianly isotopic to L

p0

= { (q, p

0

) ∈ T

^∗

T

ⁿ

} and H(L) ≥ h (resp. ≤ h) we have A (H)(p

0

) ≥ h (resp. ≤ h).

(6) We have

k→∞

lim 1

k c

+

(ϕ

^k

) = sup

p∈Rⁿ

H(p)

k→∞

lim 1

k c

−

(ϕ

^k

) = inf

p∈Rⁿ

H(p) (7) Given any measure µ on R

ⁿ

the map

ζ (H) = Z

Rⁿ

A (H)(p)dµ(p)

is a symplectic quasi-state (cf. [E-P] for the definition and proper- ties of this notion, based on earlier work by [Aarnes]). In particular we have A (H + K) = A (H) + A (K) whenever H and K Poisson-commute (i.e. { H, K } = 0).

Remarks 3.3. • In (1) the assumption could be replaced by the property

that H

1

H

2

in the sense of [V1].

• As a result of (5) if u is a smooth subsolution of the stationary Hamilton- Jacobi equation, that is H(x, p+du(x)) ≤ h then H(p) ≤ h. Similarly if u is a smooth supersolution, that is H(q, p +du(q)) ≥ h then H(p) ≥ h.

• From (5), we get the following statement: let

E

_c⁺

= { p

₀

∈ R

ⁿ

| ∃ L Hamiltonianly isotopic to L

_p0

, H (L) ≥ c }

E

_c⁻

= { p

0

∈ R

ⁿ

| ∃ L Hamiltonianly isotopic to L

p0

, H (L) ≤ c } As a result, if p ∈ E

⁺_c

∩ E

⁻_c

, we have H(p) = c.

3.2 Partial Homogenization

We here consider the case where the Hamiltonian is defined on T

^∗

T

ⁿ

× M , where M is some symplectic manifold. We shall only consider here the case where M = T

^∗

T

^m

, but the general case can be easily adapted.

Theorem 3.4 (Main theorem, partial homogenization case). Let H(x, y, q, p) be a Hamiltonian on T

^∗

T

^n+m

. Then

(1) The sequence

H

k

(x, y, q, p) = H(kx, y, q, p) γ-converges to H(y, q, p)

(2) The map

A

^x

: C

_c²

([0, 1] × T

^∗

T

^n+m

, R ) → C

⁰

( R

ⁿ

× T

^∗

T

^m

, R )

given by A

x

(H) = H extends to a projector (i.e. it satisfies A

²x

= A

x

) with Lipschitz constant 1

A

^x

: c H (T

^∗

T

^n+m

) → c H ( R

ⁿ

× T

^∗

T

^m

) where the metric on c H is γ.

(3) If H

_(q,p)

(x, y) = H(x, y, q, p), we have

A

^x

(H)(y, q, p) = A (H

(q,p)

)(y)

Remark 3.5. (1) The Hamiltonian H(y, q, p) is called the effective Hamil- tonian. In case it is smooth, its flow is given by Φ(x

₀

, y

₀

, q

₀

, p

₀

) = (x(t), y(t), q(t), p(t))

y(t) = y

0

, x(t) = Z

t

0

∂H

∂y (y

0

, q(t), p(t))dt,

˙

q(t) = ∂H

∂p (y

0

, q(t), p(t)), p(t) = ˙ − ∂H

∂q (y

0

, q(t), p(t))

(2) It is not true anymore that H depends only on the time one map of H.

It however only depends on the family of time one maps of H

_(q,p)

. (3) More generally, using (3), we may translate in our situation the prop-

erties of A stated in the first proposition. The projector A

x

is not in- variant by symplectic maps. It is however invariant by fiber-preserving hamiltonian symplectic maps: if ψ = ψ

1

× ψ

2

. in other words

A

x

(H ◦ ψ)(y, q, p) = A

x

(H)(y, ψ

2

(q, p))

3.3 Homogenized Hamilton-Jacobi equations

Our theorem has some interesting applications to generalized solutions of evolution Hamilton-Jacobi equations

(HJ)

_∂

∂t

u(t, q) + H(t, q,

_∂x^∂

u(t, x)) = 0 u(0, q) = f (q) .

where t ∈ R , q ∈ T

ⁿ

.

Smooth solutions to such equations are only defined for t less than some T

0

: solutions exhibit shocks, that is | u |

C¹([0,T]×Tⁿ,R)

blows up as T goes to T

0

.

There are essentially two types of generalized solutions for such equations : viscosity solutions (cf. [C-L]) and variational solutions (cf. [O-V]). These two solutions do not coincide in general, with one notable exception: when the Hamiltonian is convex in p.

From [L-P-V] it follows that if H is convex in p, and u

k

is the solution of (HJ

k

)

_∂

∂t

u

k

(t, q) + H(kt, kq,

_∂x^∂

u

k

(t, q)) = 0

u

k

(0, q) = f (q)

the sequence (u

k

)

k≥1

converges to u, the solution of (HJ)

_∂

∂t

u(t, q) + H(

_∂x^∂

u(t, q)) = 0 u(0, q) = f (q)

Our theorem, together with results by Humili`ere (cf. [Hu]) implies that this extends to the non convex case, provided u

_k

is the variational solution and H is given by our main theorem. We now state the more general proposition, yielding the analog of [L-P-V] when n = 0:

Proposition 3.6. Let H ∈ C

⁰

(T

^∗

T

^n+m

), f ∈ C

⁰

(T

^n+m

) and u

k

be the vari- ational solution of

_∂

∂t

u

_k

(t, x, q) + H(kx, q,

_∂x^∂

u

_k

(t, x, q)) = 0 u

k

(0, x, q) = f (x, q)

(HJP

k

) Then lim

k→+∞

u

_k

(t, q) = u(t, q) where convergence is uniform on compact time intervals and u is the variational solution of (HJP ).

_∂

∂t

u(t, x, q) + H(kx, q,

_∂x^∂

u(t, x, q)) = 0 u(0, x, q) = f (x, q)

(HJP )

More precisely, there is a sequence ε

k

going to zero, such that

| u

k

(t, x, q) − u(t, x, q) | ≤ ε

k

t

The next three sections will be devoted to the proof of our main theorem, first in the “standard case”, then in the “partial homogenization” setting.

4 Proof of the main theorem.

Let us give the reader the main steps of the proof. We denote by ϕ

^t_k

be the flow of H(k · q, p). Starting from a G.F.Q.I. of the flow ϕ

^t

= ϕ

^t₁

, we shall in the first part of subsection 4.1, construct a G.F.Q.I. of ϕ

^t_k

.

our proof will then be split in two steps

• Finding a candidate ϕ

^t

for the limit of ϕ

_k^t

• Showing that the limit of ϕ

^t_k

is indeed ϕ

^t

The first step goes along the following lines: if H is independent form q, then c(µ

_q

⊗ 1(p), ϕ

^t_k

) = H(p), so we have to prove that

k→∞

lim c(µ

q

⊗ 1(p), ϕ

^t_k

) = H(p)

exists. This is the second part of subsection 4.1 and is proved in proposition 4.7.

The second step is more delicate, and is dealt with in subsection 4.2. In fact, the formula obtained for the G.F.Q.I. of ϕ

^t_k

yields an inequality valid for any Hamiltonian map α

k→∞

lim inf c(µ, ϕ

k

α) ≤ c(µ, ϕα) proved in proposition 4.11.

We must then prove the reverse inequality, This relies in a crucial way on the main result of [V5], which implies that a 1/k-periodic Lagrangian in T

^∗

T

ⁿ

contained in the unit disc bundle, has γ norm converging to zero.

We now give the details of the proof.

4.1 Reformulating the problem and finding the ho- mogenized Hamiltonian.

First of all, we shall assume we are dealing with an autonomous Hamiltonian.

We shall see in the next section that the general case reduces to this one.

Let ϕ

^t

be the flow associated to H, and ϕ = ϕ

¹

.

Similarly let ϕ

^t_k

be the flow associated to H

k

(q, p) = H(kq, p) , and ϕ

k

= ϕ

¹_k

We first compute ϕ

_k

as a function of ϕ.

Lemma 4.1. Let ρ

k

(q, p) = (kq, p), then ϕ

k

= ρ

⁻¹_k

ϕρ

k

. Proof. The map ρ

k

is conformally symplectic, hence

dH

k

(z)ξ = dH(ρ

k

(z))dρ

k

(z)ξ = ω (X

H

(ρ

k

(z)), dρ

k

(z)ξ)

= (ρ

^∗_k

ω)(dρ

⁻¹_k

(z)X

H

(ρ

k

(z)), ξ) Since ρ

^∗_k

ω = kω, we get

X

Hk

(z) = k ((ρ

k

)

∗

X

H

) (z)

= (ρ

_k

)

_∗

(kX

_H

)(z)

The flow of kX

H

is ϕ

^kt

, hence the flow of (ρ

k

)

∗

(kX

H

) is ρ

⁻¹_k

ϕ

^kt

ρ

k

.

We are thus looking for the γ-limit of ρ

⁻¹_k

ϕ

^k

ρ

k

.

Remark : The map ρ

⁻¹_k

is not well defined on T

^∗

T

ⁿ

, so that a priori the lemma only makes sense on T

^∗

R

ⁿ

. However given a continuous path z(t) from z

0

to z

1

in T

^∗

T

ⁿ

, and u

0

such that ku

0

= z

0

, we may find a unique continuous path u(t) such that u(0) = u

0

and k.u(t) = z(t). Therefore given an isotopy ψ

^t

starting at the identity, there is an unique isotopy ψ e

^t

such that ρ

k

ψ e

^t

= ψ

t

. Moreover the map ψ e

¹

only depends on ψ

¹

and not on the choice of the Hamiltonian isotopy.

Note also that we may replace ϕ = ϕ

¹

by ϕ

^1/r

for some fixed integer r.

Indeed, if ρ

⁻¹_k

ϕ

^k/r

ρ

k

c-converges to ψ, we have that ρ

⁻¹_k

ϕ

^k

ρ

_k

= ρ

_r

ρ

⁻¹_kr

ϕ

^kr/r

ρ

_kr

ρ

⁻¹_r

c-converges to ρ

r

ψρ

⁻¹_r

. If our theorem is proved for ϕ

^1/r

, ψ will be generated by a Hamiltonian depending only on the p variable. We easily check that in this case

ρ

r

ψρ

⁻¹_r

= ψ

^r

. In other words, ρ

⁻¹_k

ϕ

^k

ρ

_k

converges to ψ

^r

.

We assume in the sequel that ϕ is C

¹

close to the identity, so that it lifts to a Hamiltonian diffeomorphism of T

^∗

R

ⁿ

C

¹

close to the identity, ϕ. The e map ϕ e has a generating function

S(Q, p) = b h p, Q i + S(Q, p)

where S is defined on T

^∗

T

ⁿ

, and S defines ϕ e by the relation e

ϕ

Q + ∂S

∂p (Q, p), p

=

Q, p + ∂S

∂Q (Q, p)

or else

e ϕ ∂ S b

∂p (Q, p), p

!

= Q, ∂ S b

∂Q (Q, p),

!

This means that the graph of ϕ e , Γ(ϕ), in T

^∗

R

ⁿ

× T

^∗

R

ⁿ

≃ T

^∗

∆

_R²ⁿ

, has the compact supported generating function S(q, p) defined on T

^∗

T

ⁿ

.

In other words, if ϕ(q, p) = (Q, P e ) we have P − p =

_∂Q^∂S

(Q, p)

q − Q =

^∂S_∂p

(Q, p)

We now give the composition law for generating functions, due to Chekanov

(cf. [Che])

Lemma 4.2. Let ϕ

1

, ϕ

2

be Hamiltonian maps having S

1

, S

2

as generating functions. Then ϕ

₁

◦ ϕ

₂

has the generating function

S(q

1

, p

2

; q

2

, p

1

) = S

1

(q

1

, p

1

) + S

2

(q

2

, p

2

) + h p

1

, q

1

i + h p

2

, q

2

i − h p

1

, q

2

i − h p

2

, q

1

i .

= S

1

(q

1

, p

1

) + S

2

(q

2

, p

2

i + h p

1

− p

2

, q

1

− q

2

i . Note that if we set p

2

= p

1

− v, q

2

= q

1

+ u, we have

S(q

1

, p

2

; q

1

+ u, p

2

+ v) = S

1

(q

1

, p

2

+ v) + S

2

(q

1

+ u, p

2

) − h v, u i so that S is a generating function quadratic at infinity.

Proof. It is a simple computation. The lemma claims that S(q b

1

, p

2

; q

2

, p

1

) = S b

1

(q

1

, p

1

) + S b

2

(q

2

, p

2

) − h p

1

, q

2

i so we must prove that ϕ

₁

◦ ϕ

₂

maps

∂ S b

∂p

2

(q

1

, p

2

; q

2

, p

1

), p

2

!

= ∂ S b

2

∂p

2

(q

2

, p

2

), p

2

!

to

q

₁

, ∂ S b

∂q

1

(q

₁

, p

₂

; q

₂

, p

₁

)

!

= q

₁

, ∂ S b

₁

∂q

1

(q

₁

, p

₁

)

!

where (q

₁

, p

₁

, q

₂

, p

₂

) are constrained by

 

 

 

 

 



∂ S b

∂p

1

(q

1

, p

2

; q

2

, p

1

) = 0 i.e. q

2

= ∂ S b

1

∂p

1

(q

1

, p

1

)

∂ S b

∂q

2

(q

₁

, p

₂

, q

₂

, p

₁

) = 0 i.e. p

₁

= ∂ S b

2

∂q

2

(q

₂

, p

₂

) These last two equations are equivalent to

ϕ

₂

∂ S b

₂

∂p

2

(q

₂

, p

₂

), p

₂

!

= q

₂

, ∂ S b

₂

∂q

2

(q

₂

, p

₂

)

!

= (q

₂

, p

₁

)

and

ϕ

1

(q

2

, p

1

) = ϕ

1

∂ S b

₁

∂p

1

(q

1

, p

1

), p

1

!

= q

1

, ∂ S b

₁

∂q

1

(q

1

, p

1

)

!

and thus

ϕ

1

◦ ϕ

2

∂ S b

∂p

2

(q

1

, p

2

; q

2

p

1

), p

2

!

= q

1

, ∂ S b

∂q

1

(q

1

, p

2

; q

2

, p

1

)

!

where (q

1

, p

2

; q

2

, p

1

) are constrained by

∂ S b

∂q

2

(q

1

, p

2

; q

2

, p

1

) = ∂ S b

∂p

1

(q

1

, p

2

; q

2

, p

1

) = 0

More generally, we get

Lemma 4.3. The map ϕ e

^k

is generated by

S b

k

(q

1

, p

k

; p

1

, q

2

, p

2

, · · · , q

k−1

, p

k−1

, q

k

) =

Σ

_k

(q

₁

, p

_k

; p

₁

, q

₂

, p

₂

, · · · , q

_k−1

, p

_k−1

, q

_k

) + Q

_k

(q

₁

, p

_k

; p

₁

, q

₂

, p

₂

, · · · , q

_k−1

, p

_k−1

, q

_k

) where

Σ

_k

(q

₁

, p

_k

; p

₁

, q

₂

, p

₂

, · · · , q

_k−1

, p

_k−1

, q

_k

) = X

k

j=1

S(q

_j

, p

_j

) and

Q

_k

(q

₁

, p

_k

, p

₁

, q

₂

, · · · , q

_k−1

, p

_k−1

, q

_k

) = X

k−1

j=1

h p

_j

− p

_j+1

, q

_j

− q

_j+1

i + h p

_k

, q

₁

i Proof. The proof follows immediately by induction from the previous lemma.

Note that Q

_k

could be rewritten as

Q

k

(q

1

, p

k

; p

1

, q

2

, · · · , q

k−1

, p

k−1

, q

k

) = X

k−1

j=1

h p

j

, q

j

− q

j+1

i + h p

k

, q

k

i .

Again, Σ

k

is defined on (T

^∗

T

ⁿ

)

^k

, while Q

k

is defined on (T

^∗

R

ⁿ

)

^k

. Finally we

have

Lemma 4.4. Let ϕ be generated by S(q, p), then ϕ

k

= ρ

⁻¹_k

ϕ

^k

ρ

k

is generated by F b

k

given by

F b

_k

(q

₁

, p

_k

; p

₁

, · · · , q

_k−1

, p

_k−1

, q

_k

) = 1

k Σ

_k

(kq

₁

, p

_k

; p

₁

, · · · , kq

_k−1

, p

_k−1

, kq

_k

) + Q

_k

(q

₁

, p

_k

; p

₁

, · · · , q

_k−1

, p

_k−1

, q

_k

) Proof. Indeed if S(q, p; ξ) is a generating function for ψ, we have that ρ

⁻¹_k

ψρ

k

is generated by

¹_k

S(kq, p; ξ).

Thus in our case, we expect the generating function G b

_k

(q

₁

, p

_k

; p

₁

, q

₂

, · · · , q

_k−1

, p

_k−1

, q

_k

) = 1

k Σ b

k

(kq

1

, p

k

; p

1

, q

2

, · · · , q

k−1

, p

k−1

, q

k

) + 1

k Q

k

(kq

1

, p

k

; p

1

, q

2

, · · · , q

k−1

, p

k−1

, q

k

) .

But the fiber preserving change of variable q

j

→ kq

j

(j ≥ 2) sends G b

k

to F

k

(q

1

, p

k

; p

1

, q

2

, · · · , q

k−1

, p

k−1

, q

k

) =

1

k Σ b

k

(kq

1

, p

k

; p

1

, kq

2

, · · · , kq

k−1

, p

k−1

, q

k

) + 1

k Q

_k

(kq

₁

, p

_k

; p

₁

, kq

₂

, · · · , kq

_k−1

, p

_k−1

, q

_k

) . It is easy to check that the last term is equal to

Q

k

(q

1

, p

k

; p

1

, q

2

, · · · , q

k−1

, p

k−1

, p

k

) .

We now set

F

k

(q

1

, p

k

; p

1

, q

2

, · · · , q

k−1

, p

k−1

, p

k

) = F b

k

(q

1

, p

k

; p

1

, q

2

, · · · , q

k−1

, p

k−1

, p

k

) − h p

k

, q

1

i and to simplify our notations

x = q

1

, y = p

k

, ξ = (p

1

, q

2

, · · · , q

k−1

, p

k−1

, p

k

) . thus

F

k

(x, y; ξ) = X

k

j=1

S(q

j

, p

j

) + Q

k

(x, y, ξ) − h y, x i

Definition 4.5. We set h

k

(y) = c(µ

x

, F

k,y

) where F

k,y

= F

k

(x, y; ξ). We will also denote this function as c(µ

_x

⊗ 1(y), F

_k

).

Remark 4.6. As long as we write c(µ

x

⊗ 1(y), S) for a generating function S, there is no ambiguity. However, if Λ is the Lagrangian associated to S, and we write an expression like c(µ, (Λ)

y

), one should be careful since S is only defined up to a constant, and this constant yields a coherent choice of a G.F.Q.I. for (Λ)

_y

for each y, so that the c(µ, Λ

_y

) are well-defined up to the same constant for all values of the parameter y, and not up to a function of y as one could expect.

Clearly we take a G.F.Q.I. S for L and then (L)

_y

has G.F.Q.I. S

_y

= S(y, • ).

Proposition 4.7. The sequence (h

_k

) is a precompact sequence for the C

⁰

topology.

The proposition will follow from Ascoli-Arzela’s theorem once we prove the following

Lemma 4.8. The sequence of (h

k

) is equicontinuous.

Proof. Indeed let ϕ e be the lift of ϕ

k

= ρ

⁻¹_k

ϕ

^k

ρ

k

to T

^∗

R

ⁿ

. It has support in some tube

T

_A^∗

R

ⁿ

= { (q, p) ∈ T

^∗

R

ⁿ

| | p | ≤ A } . Now for each p there exists q(p), ξ(p) such that

∂F

_k

∂x (q(p), p; ξ(p)) = 0, ∂F

_k

∂ξ (q(p), p, ξ(p)) = 0 F

k

(q(p), p, ξ(p)) = h

k

(p) .

Moreover, for generic ϕ, the map p → (q(p), ξ(p)) is smooth on the comple- ment of some codimension 1 set. Thus for p in this complement,

dh

k

(p) = ∂

∂y F

k

(x(p), p, ξ(p))

= q(p) − Q

k

(q(p), p) where Q

_k

is defined by

e

ϕ

k

(q, p) = (Q

k

(q, p), P

k

(q, p)) .

The quantity q(p) − Q

_k

(q(p), p) can be estimated as follows : the first coor- dinate of the flow ϕ e

^t_k

satisfies

˙

q

k

(t) = ∂H

∂p (kq

k

(t), p

k

(t))

hence | q ˙

k

(t) | is bounded by C = sup n

^∂H∂p

(q, p)

| (q, p) ∈ T

^∗

T

ⁿ

o hence

| q − Q

k

(q, p) | ≤ C. From this we get the inequality

| dh

k

(p) | = ∂

∂p F

k

(q(p), p, ξ(p)) ≤ C and since h

k

is continuous, it is C-Lipschitz.

From Ascoli-Arzela and the above lemma, we see that the sequence h

k

is relatively compact.

We will then need to prove that

Lemma 4.9. If a subsequence of ρ

⁻¹_k

ϕ

^k

ρ

k

has a limit ϕ, then the sequence itself converges to this limit.

Proof. We first claim that

(1) γ(ϕ

^k

ψ

^k

) ≤ kγ(ϕψ)

(2) γ(ρ

⁻¹_k

ϕρ

_k

) = 1

k γ(ϕ) . Indeed, we may write

ϕ

^k

ψ

^k

= ϕψ(ψ

⁻¹

(ϕψ)ψ)(ψ

⁻²

(ϕψ)ψ

²

)...ψ

^−(k−1)

(ϕψ)ψ

^k−1

Since each factor is conjugate to ϕψ, and we have k factors, property (1) follows immediately. Property (2) follows from the scaling property of γ by conformal conjugation (see 1).

Now let k large enough, so that γ(ρ

⁻¹_k

ϕ

^k

ρ

k

, ϕ) is less than ε. Let n = kq + r. Note first that if r = 0,

γ(ρ

⁻¹_kd

ϕ

^kd

ρ

kd

, ϕ) = γ(ρ

⁻¹_d

(ρ

⁻¹_k

ϕ

^k

ρ

k

)

^d

ρ

d

(ρ

⁻¹_d

ϕ

^−d

ρ

d

)) since

ϕ = ρ

⁻¹_d

ϕ

^d

ρ

_d

But using (2), we get

γ(ρ

⁻¹_d

(ρ

⁻¹_k

ϕ

^k

ρ

_k

)

^d

ρ

_d

(ρ

⁻¹_d

ϕ

^−d

ρ

_d

)) = γ(ρ

⁻¹_d

(ρ

⁻¹_k

ϕ

^k

ρ

_k

)

^d

ϕ

^−d

)ρ

_d

)) ≤ 1

d γ((ρ

⁻¹_k

ϕ

^k

ρ

_k

)

^d

ϕ

^−d

) ≤ 1

d dγ (ρ

⁻¹_k

ϕ

^k

ρ

k

ϕ

⁻¹

) ≤ ε

Then we claim that

q→∞

lim γ (ρ

⁻¹_kq+r

ϕ

^kq+r

ρ

kq+r

, ρ

⁻¹_kq

ϕ

^kq

ρ

kq

) = 0

Indeed, working in T

^∗

R

ⁿ

, we may write ρ

kq+r

ρ

⁻¹_kq

= ρ

1+(r/kq)

and ϕ

^kq+r

ϕ

^−kq

= ϕ

^r

thus

γ(ρ

⁻¹_kq+r

ϕ

^kq+r

ρ

kq+r

, ρ

⁻¹_kq

ϕ

^kq

ρ

kq

) = γ (ρ

⁻¹_1+(r/kq)

(ρ

⁻¹_kq

ϕ

^kq

ρ

kq

)(ρ

⁻¹_kq

ϕ

^r

ρ

kq

)ρ

1+(r/kq)

, ρ

⁻¹_kq

ϕ

^kq

ρ

kq

) = γ(ρ

⁻¹_1+(r/kq)

(ρ

⁻¹_kq

ϕ

^kq

ρ

kq

)(ρ

⁻¹_kq

ϕ

^r

ρ

kq

)ρ

1+(r/kq)

ρ

⁻¹_kq

ϕ

^−kq

ρ

kq

)

Now we use the fact we proved earlier, that γ(ρ

⁻¹_kq

ϕ

^kq

ρ

kq

, ϕ) ≤ ε and that

γ(ρ

⁻¹_kq

ϕ

^r

ρ

kq

) ≤ 1

kq γ(ϕ

^r

) ≤ ε for q large enough.

We use the fact thatϕ and ρ

s

satisfy ρ

⁻¹_s

ϕ = ϕ

^s

ρ

s

to infer the next estimate of the above quantity

γ(ρ

⁻¹_1+(r/kq)

ϕψ

1

ρ

1+(r/kq)

ψ

2

ϕ

⁻¹

) = γ(ϕ

^r/kq

ρ

⁻¹_1+(r/kq)

ψ

1

ρ

1+(r/kq)

ψ

2

)

where γ (ψ

j

) ≤ ε. The above is then small as soon as kq is large enough.

We thus proved that for q large enough,

γ(ρ

⁻¹_kq+r

ϕ

^kq+r

ρ

kq+r

, ρ

⁻¹_kq

ϕ

^kq

ρ

kq

)

is close to zero, Since we proved earlier that γ(ρ

⁻¹_kq

ϕ

^kq

ρ

kq

) ≤ ε this proves that the limit of ρ

⁻¹_k

ϕ

^k

ρ

k

is indeed ϕ.

4.2 Concluding the proof of the main theorem.

The last section gave us a function h

∞

(p) limit of some subsequence h

kν

(p).

Since h

∞

is continuous, according to Humili`ere (cf. prop.2.11) it has a flow in H b (T

^∗

T

ⁿ

). We denote by ϕ

^t

this flow.

Proposition 4.10. The map ϕ is the limit of ϕ

k

= ρ

⁻¹_k

ϕ

^k

ρ

k

: we have

k→+∞

lim γ (ϕ

_k

, ϕ) = 0 .

This will be based on the following two propositions

Proposition 4.11. For any α in H b (T

^∗

T

ⁿ

), there exists a sequence ℓ

ν

such that

ν→∞

lim c(µ, ϕ

ℓ^ν

α) ≤ c(µ, ϕα)

Proposition 4.12. Consider a subsequence of (ϕ

k^ν

) such that lim

ν

c(µ ⊗ 1(p), ϕ

k^ν

) = lim

ν

h

k^ν

(p) = h

∞

(y) Then we have

lim

ν

c(µ ⊗ 1(p), ϕ

⁻¹_kν

) = − h

∞

(p)

Remark 4.13. Note that this means that if we define H as in proposition 4.7, the operator A satisfies A ( − H) = −A (H). This is typically a statement that does not hold in the case of viscosity solutions, since if u(t, x) is a vis- cosity solution associated to H, u( − t, x) is not in general a viscosity solution associated to − H.

Proof that Proposition 4.11 and 4.12 imply proposition 4.10. Indeed take α = ϕ

⁻¹

, where ϕ is the limit associated by the previous subsection to some sub- sequence (k

ν

)

ν≥1

. We get

lim

ν

c(µ, ϕ

ℓ^ν

ϕ

⁻¹

) ≤ c(µ, Id) = 0.

and since for any ψ, c(µ, ψ) ≥ 0 we get,

lim

ν

c(µ, ϕ

ℓ^ν

ϕ

⁻¹

) = 0

Now we must prove lim

ν

c(1, ϕ

ν·k^ν

ϕ

⁻¹

) = 0, and it is enough to show that lim

ν

c(1, ϕ

ℓ^ν

α)) ≥ c(1, ϕα)

for any α in H b (T

^∗

T

ⁿ

).

But

c(1, ϕ

ν·kν

α) = − c(µ, α

⁻¹

ϕ

⁻¹_ℓν

) = − c(µ, ϕ

⁻¹_ℓν

α

⁻¹

) .

We may then apply proposition 4.11 to the sequence (ϕ

⁻¹_kν

) since according to proposition 4.12

lim

k

c(µ ⊗ 1(p), ϕ

⁻¹_ℓν

) = − h

∞

(p)

and − h

∞

(p) has flow ϕ

⁻¹

in the completion H b (T

^∗

T

ⁿ

).

As a result

lim

k

c(1, ϕ

_ℓν

α) = − lim

k

c(µ, ϕ

⁻¹_ℓν

α

⁻¹

)

≥ − c(µ, ϕ

⁻¹

α

⁻¹

) = c(1, ϕα) .

We thus proved the following statement: if c(µ

_x

⊗ 1(p), ϕ

_kν

) converges to h

∞

, then ϕ

ℓ^ν

converges to ϕ. Now assume there are two subsequences, ϕ

k^ν

, ϕ

l^ν

such that c(µ

x

⊗ 1(p), ϕ

k^ν

) converges to h

∞

, while c(µ

x

⊗ 1(p), ϕ

l^ν

) converges to k

_∞

. Then we find subsequences of (ϕ

_k

) converging to ϕ and ψ (where ϕ is the flow of h

∞

while ψ is the flow of ψ).

But according to lemma 4.9, two converging subsequences of (ϕ

k

) must have the same limit, thus h

∞

= k

∞

.

This concludes our proof of 4.10, modulo the proof of 4.11 and 4.12.

Proof of proposition 4.11. First of all, if S(x, y, η) is a G.F.Q.I. of α, ϕ

k

α has the G.F.Q.I.

Φ

k

(u, v; x, y, η, ξ) =

S(x, v; η) + F

_k

(u, y; ξ) + h y − x, v − u i

Note that for each y, there is a cycle C(y) homologous to T

ⁿ

× E

_k⁻

(x lives in T

ⁿ

, ξ in E

k

, E

_k⁻

is the negative eigenspace for F

k

) such that

F

k

(y, C(y)) ≤ h

k

(y) + ε .

(we denote by (y, C(y)) the set of (x, y, ξ) such that (x, ξ ) ∈ C(y)). Unfor- tunately we may not get such an estimate if we simultaneously require that C(y) is to depend continuously on y. However, we may assume the above estimate holds outside of U

2δ

where U

δ

is a δ-neighbourhood of some grid in ( R

ⁿ

)

^∗

(see figure 1), while inside U

δ

, F

k

(y, C(y)) ≤ a for some constant a.

Indeed we have

Lemma 4.14. Let F (u, x) be a smooth function on V × X such that there exists C(u) ∈ H

∗

(X) with F (u, C(u)) ≤ f(u). Then for any subset U in V , such that each connected component of V − U has diameter less than ε, there exists a continuous map u −→ C(u) e and a constant a, such that

F (u, C(u)) e ≤ f (u) + aχ

U

(u)

Thus F

k

(y, C(y)) ≤ h

k

(y) + aχ

^δ

(y) + ε where χ

^δ

is a smooth function equal to one on U

δ

and to zero outside U

2δ

, a is some constant, and ε is arbitrarily small.

Assume first U

δ

is empty (i.e. χ

^δ

= 0).

Set

Φ

k

(u, v; x, y, η) = S(x, v; η) + h

k

(y) + h y − v, u − x i

defined on T

_(u,v)ⁿ

× R

ⁿ

x

× ( R

ⁿ

y

)

^∗

× N (N is the vector spaces where η lives).

Let Γ be a cycle in T

_(u,v)ⁿ

× ( R

_x

)

ⁿ

× ( R

ⁿ

y

)

^∗

× N in the homology class of T

_(u,v)ⁿ

× ∆

_x,y

× N

⁻

(∆

_x,y

is the diagonal in R

ⁿ

x

× ( R

ⁿ

y

)

^∗

and the negative eigenspace of h y, − x i ).

We choose Γ such that Φ

k

(Γ) ≤ c(µ, Φ

k

) + ε = c(µ, ϕ

_k

α) + ε, which is possible by definition.

Let now Γ ×

Y

C be the cycle

Γ ×

Y

C = { (u, v, x, y, ξ, η) | (u, v, x, y, η) ∈ Γ, (u, ξ) ∈ C(y) } . We claim that

Φ

k

(Γ ×

^Y

C) ≤ Φ

k

(Γ) and Γ ×

^Y

C is a cycle in the homology class of

T

_(u,v)ⁿ

× ∆

x,y

× E

_k⁻

× N

₁⁻

× N

₂⁻

so that

c(µ, ϕ

k

α) = c(µ, Φ

k

) ≤ Φ

k

(Γ) ≤ c(µ, Φ

k

) + ε ≤ c(µ, ϕ

_k

α) + ε Let us now try to establish the inequality in the general case,

Let k

ν

be a sequence such that c(µ

x

⊗ 1(y), ϕ

k^ν

) converges to h

∞

(y). We replace Φ

k

by Φ

ℓ,k

where F

k

is replaced by the explicit formula for F

ℓk

given by lemma 4.4.

Φ

ℓ,k

(u, v; x, y, ξ, η) = S(x

1

, v, η)+ 1 ℓ

X

ℓ j=1

F

k

(ℓx

j

, y

j

, ξ

j

)+Q

ℓ

(x, y)+ h y

ℓ

− v, u − x

1

i We may now “spread our error” aχ

^δ

(y) by translating it. More precisely, let us choose different χ

^δ_j

(y) so that

F

k

(y, C

j

(y)) ≤ h

k

(y) + a

k

χ

^δ_j

(y) + ε

and such that the supports, U

_j^δ

, of the χ

^δ_j

are so chosen that the intersection of (n + 2) distinct U

_j^δ

is empty.

Consider

Φ

ℓ,k

(u, v ; x, y, η) = S(x

1

, v; η) + 1

ℓ X

ℓ

j=1

h

k

(y

j

) + a

k

χ

^δ_j

(y

j

)

+ Q

ℓ

(x, y) + h y

ℓ

− v, u − x

1

i

Figure 1: The sets U

_δ^j

Then, let Γ be a cycle in the same homology class as above, such that Φ

ℓ,k

(Γ) ≤ c(µ, Φ

ℓ,k

) + ε

and

Γ ×

^Y

C =

(u, v; x, y, ξ, η) | (u, v, x, y, η) ∈ Γ, (ℓx

j

, ξ

j

) ∈ C

j

(y

j

) . Now Γ ×

^Y

C belongs to the suitable homology class, and we may thus infer that

c(µ, ϕ

k

α) = c(µ, Φ

ℓ,k

) ≤ Φ

ℓ,k

(Γ ×

Y

C) and Φ

ℓ,k

(Γ ×

Y

C) ≤ Φ

ℓ,k

(Γ). We may conclude that

c(µ, Φ

_ℓ,k

) ≤ c(µ, Φ

_ℓ,k

) + 2ε Finally, we must show

Lemma 4.15. We have

c(µ, Φ

ℓ,k

) ≤ c(µ, Φ

k

) + A

k

ℓ

Proof. Indeed Φ

ℓ,k

is the generating function of ψ

k,δ,ℓ

α where ψ

k,δ,ℓ

= ρ

⁻¹_ℓ

ψ

¹_k,δ

◦ · · · ◦ ψ

_k,δ^ℓ

ρ

ℓ

where ψ

_k,δ^j

is the time one flow of h

k

(y) + a

k

χ

^δ_j

(y).

But because these flows commute, we have that ψ

k,δ,ℓ

is the time one flow of

K

k,δ,ℓ

(y) = 1 ℓ

X

ℓ j=1

h

k

(y) + a

k

χ

^j_j

(y)

! .

Now since (n+2) sets U

_δ^j

have empty intersection, we have that P

ℓ

j=1

χ

^δ_j

(y) ≤ (n + 2) hence

| K

k,δ,ℓ

(y) − h

k

(y) | ≤ A

k

ℓ As a result

γ(ψ

k,δ,ℓ

, ψ

k

) ≤ A

k

ℓ where ψ

k

is the time one flow of h

k

(y) hence

γ (ψ

k,δ,ℓ

α, ψ

k

α)

≤ A

k

ℓ and

c(µ, Φ

ℓ,k

) ≤ c(µ, ψ

k,δ,ℓ

α)

≤ c(µ, ψ

k

α) + a

_k

ℓ

≤ c(µ, ϕ

_k

α) + A

k

ℓ

Since by assumption, as ν goes to infinity along a subsequence ψ

k^ν

goes to ϕ we get

c(µ, ϕ

_kℓ

α) = c(µ, Φ

_ℓ,k

) ≤ c(µ, Φ

_ℓ,k

) + ε ≤ c(µ, Φ

_k

) + ε + a

_k

ℓ

≤ c(µ, ϕα) + A

k

ℓ Taking ℓ large enough, we see that

lim

ν

c(µ, ϕ

ℓ^ν·k^ν

α)

≤ c(µ, ϕα)

as announced. This concludes the proof of Proposition 4.11.

Remark 4.16. It is important to notice that here Φ

ℓ,k

= S(x

1

, v; η) + 1

ℓ X

ℓ

j=1

h

k

(y

j

) + aχ

^δ_j

(y

j

)

+ Q

ℓ

(x, y) + h y

ℓ

− v, u − x

1

i cannot be bounded from above by

S(x

1

, v; η) + 1 ℓ

X

ℓ j=1

h

k

(y

j

) + Q

ℓ

(x, y) + h y

ℓ

− v, u − x

1

i + a ℓ

as it is obvious by choosing (y

1

, ..., y

ℓ

) such that each y

j

is in U

_j^δ

. the above

proof would not hold if we replace χ

^δ_j

(y) by an analogous function χ

^δ_j

(q, p).

Proof of proposition 5.3. Let Γ

ϕ

be the graph of ϕ in coordinates

Γ

_ϕ

= { (X(x, y), x, y − X(x, y), X (x, y) − x) | ϕ(x, y) = (X(x, y ), Y (x, y)) } Then the reduction of Γ

ϕ

at y = y

0

is (Γ

ϕ

)

y0

= { (X(x, y), y

0

− Y (x, y

0

)) } that is L

y⁰

− ϕ(L

y⁰

) where

L

y0

= { (x, y

0

) | x ∈ T

ⁿ

}

Now c(µ

x

⊗ 1(y), Γ

ϕ

) = c(µ, (Γ

ϕ

)

y

) = c(µ, L

y

− ϕ(L

y

)).

Remark 4.17. We refer to remark 4.6 and remind once again the reader of the extra care that has to be taken when using such expressions as c(µ, Λ

y

).

Here c(µ, (Γ

_ϕ

)

_y

)) is well defined if we choose a G.F.Q.I. for Γ

_ϕ

.

If ϕ = ϕ

k

= ρ

⁻¹_k

ϕ

^k

ρ

k

, then ϕ

k

is 1/k-periodic, hence so is L

y

− ϕ

k

(L

y

).

Lemma 4.18. The quantity γ(L

y

− ϕ

k

(L

y

)) goes to 0 as k goes to infinity.

Proof. Note that L

y

is not exact, but if τ

y

is the translation by the vector y in the p direction

L

y

= τ

y

(0

Tⁿ

) ϕ

k

(L

y

) = ϕ

k

τ

k

(0

Tⁿ

) and we identify

γ(L

y

− ϕ

k

(L

y

)) to

γ 0

Tⁿ

− (τ

_y⁻¹

ϕ

k

◦ τ

y

)(0

Tⁿ

)

= γ(τ

_y⁻¹

ϕ

k

τ

y

(0

Tⁿ

))

Moreover (τ

_y⁻¹

ϕ

k

τ

y

)(0

Tⁿ

) is Hamiltonianly isotopic to the zero section and invariant by any translation in the x direction by a vector in

_k¹

Z

ⁿ

. More precisely the Hamiltonian isotopy τ

_y⁻¹

ϕ

k

τ

y

)(0

Tⁿ

) is invariant by such trans- lation.

Thus γ (τ

_y⁻¹

ϕ

k

τ

y

(0

Tⁿ

)) is the γ invariant of some Lagrangian contained in the product T

₁^∗

T

_kⁿ

(where T

_kⁿ

= ( R /

_k¹

Z )

ⁿ

), Hamiltonianly isotopic (in T

₁^∗

T

_kⁿ

) to the zero section. Since

T

₁^∗

T

_kⁿ

≃ T

_1/k^∗

T

ⁿ

= { (q, p) | x ∈ T

ⁿ

, | p | ≤ 1 k } we get, according to the main result in [V5], that

γ(τ

_y⁻¹

ϕ

_k

τ

_y

(0

_Tⁿ

)) ≤ 0(1/k)

As a result, c(µ, L

y

− ϕ

k

(L

y

)) = − c(1, ϕ

k

(L

y

) − L

y

) = − c(1, L

y

− ϕ

⁻¹_k

(L

y

)) This last quantity differs from

− c(µ, L

y

− ϕ

⁻¹_k

(L

y

)) by

γ(L

y

− ϕ

⁻¹_k

(L

y

)) = γ(ϕ

k

(L

y

) − L

y

) which goes to zero as k goes to infinity.

Hence

c(µ, L

y

− ϕ

⁻¹_k

(L

y

)) = c(µ, ϕ

k

(L

y

) − L

y

) = − c(1, L

y

− ϕ

k

(L

y

)) ≃ − c(µ, L

y

− ϕ

k

(L

y

)) In other words, denoting h

k

the number c(µ

x

⊗ 1(y), ϕ

⁻¹_k

), we proved

h

_k

(y) = c(µ

_x

⊗ 1(y), ϕ

⁻¹_k

) = c(µ ⊗ 1(y), L

_y

− ϕ

⁻¹_k

(L

_y

)) =

− c(µ ⊗ 1(y), L

y

− ϕ

⁻¹_k

(L

y

)) + 0( 1

k ) = − h

k

(y) + 0( 1 k )

We thus showed that some subsequence of ϕ

k

converges to ϕ. On the other hand, we proved in lemma 4.9 that this implies convergence of the sequence itself to ϕ. Wa may finally conclude that ϕ

k

converges to ϕ, and thus that the sequence H(kq, p) converges to H for the γ metric, which proves the first statement of the main theorem.

End of the proof of theorem 3.1. Assertion (2) follows from the fact that ϕ

¹

determines H, and that

ϕ

¹

= lim

k→∞

ρ

⁻¹_k

ϕ

^k

ρ

k

which only depends on ϕ

¹

.

We finally prove assertion (3).

| h

k,1

(y) − h

k,2

(y) | ≤ | c(µ

x

⊗ 1(y), ρ

⁻¹_k

ϕ

^k₁

ρ

k

) − c(µ

x

⊗ 1(y), ρ

⁻¹_k

ϕ

^k₂

ρ

k

) | ≤ γ (ρ

⁻¹_k

ϕ

^k₁

ρ

k

)

⁻¹

◦ ρ

⁻¹_k

ϕ

^k₂

ρ

k

≤ γ(ρ

⁻¹_k

ϕ

^−k₁

ϕ

^k₂

ρ

k

) ≤ 1

k γ(ϕ

^−k₁

ϕ

^k₂

) ≤ γ(ϕ

⁻¹₁

ϕ

2

) Therefore A is Lipschitz, with Lipschitz constant one, for the norms γ and C

⁰

and thus extends to a Lipschitz map from H b (T

^∗

T

ⁿ

) to C

⁰

( R

ⁿ

, R ).

Since is H only depends on p, then H = H, we get that A is a projector.

This concludes our proof of theorem 3.1.