APENDICES E INFORMACION DE SOPORTE 3 Cómo elaborar un Plan de Negocio.

In this subsection we use the asymptotic formula of Theorem 3.5.1 to construct an analytic invariant known as Stokes constant that measures the splitting distance of the complex invariant manifolds parametrised by Γ±. This constant is also related to the Stokes phenomenon where two difference analytic functions which possess a common asymptotic expansion in a common region differ by an exponentially small term. The Stokes constant is the normalized amplitude of this exponentially small term. In order to define this invariant, let us first prove two technical Lemmas which we will use later on. Let ∆(ϕ, τ) =Γ+(ϕ, τ)−Γ−(ϕ, τ).

Lemma 3.5.1. For everyv∈C4 _{we have,}

Ω(Θ±_∗, v) = lim

Im(τ)→±∞Ω(∆(ϕ, τ),U(ϕ, τ)v)e

∓i(τ−ϕ)_,

where the convergence of the limit in the right hand side is uniform with respect to ϕ_∈Sh.

Proof. According to Theorem 3.5.1 and Remark 3.5.1.1 we have the following asymp- totic formula,

valid inSh×D1r,± forµ0 ∈(0,1) very small, r >0 sufficiently large. Now taking into account thatU is a normalized fundamental matrix and formula (3.81) we get at once,

Ω(∆(ϕ, τ),U(ϕ, τ)v)e∓i(τ−ϕ) = Ω(U(ϕ, τ)Θ±_∗,U(ϕ, τ)v) +Oe±(1−µ0)i(τ−ϕ), = Ω(Θ±_∗, v) +Oe±(1−µ0)i(τ−ϕ)_.

which proves the desired formula by taking the limit as Im(τ) _{→ ±∞}. Moreover it is clear that the convergence is uniform with respect toϕ∈Sh.

Lemma 3.5.2. The following limits exist, are independent ofϕand the convergence is uniform inSh, Θ±₀ := lim Im(τ)→±∞Ω(∆(ϕ, τ), ∂ϕΓ −_{(ϕ, τ))e}∓i(τ−ϕ) _{<∞. (3.82)} Moreover, 1. Θ±₀ =₋ lim Im(τ)→±∞Ω(∆(ϕ, τ), ∂τΓ −_{(ϕ, τ))e}∓i(τ−ϕ)_,

2. If H is real analytic then,

Θ+₀ =      −Θ−₀ if η >0, Θ−₀ if η <0.

3. For any other solutions Γ˜± _∈ X1(Sh ×Dr±˜) of equation (3.4) such that Γ˜± ≍ ˆ

˜

Γ where Γˆ˜ _∈ τ−1T4C[[τ−1]] is a formal solution of equation (3.4) we have the

following relationΘ˜±₀ = Θ±₀e±i(τ0−ϕ0) _{for some(ϕ}

0, τ0)∈C2where the definition of Θ˜±₀ is analogous to (3.82) for the parametrisations Γ˜±.

Proof. That the limits (3.82) exist and are uniform with respect to ϕ follows from the previous Lemma withv= (1,0,0,0). Now let us prove that

Θ−₀ =− lim

Im(τ)→−∞Ω(∆(ϕ, τ), ∂τΓ

−_{(ϕ, τ))e}i(τ−ϕ)_, (the+case being completely analogous). First note that (3.81) implies,

Now taking into account thatH(Γ±(ϕ, τ)) = 0we get, lim Im(τ)→−∞∇H(Γ −_{(ϕ, τ))∆(ϕ, τ)e}i(τ−ϕ) _{= 0. (3.83)} Moreover, ∇H(Γ−)∆ = Ω(XH(Γ−),∆) = Ω(DΓ−,∆) =− Ω(∆, ∂ϕΓ−) + Ω(∆, ∂τΓ−). Thus, (3.83) yields, lim Im(τ)→−∞ Ω(∆(ϕ, τ), ∂ϕΓ −_{(ϕ, τ)) + Ω(∆(ϕ, τ), ∂} τΓ−(ϕ, τ))ei(τ−ϕ) = 0 which proves the desired equality.

Now suppose thatH is real analytic andη >0. Let us prove thatΘ−₀ =−Θ+₀. Since Θ−₀ is defined by a limit as Im(τ) _{→ −∞} we can take a sequence τn = −iσn whereσn is any real sequence such thatσn→+∞ asn→+∞. Then,

Θ−₀ = lim

n→+∞Ω(∆(0,−iσn), ∂ϕΓ

−_(0,−iσ

n))eσn.

Now it follows from Remark 3.2.2.2 that∆(0,−iσn) = ∆(π, iσn)and∂ϕΓ−(0,−iσn) = ∂ϕΓ−(π, iσn). Thus, Θ−₀ = lim n→+∞Ω(∆(0,−iσn), ∂ϕΓ −_(0,−iσ n))eσn = lim n→+∞Ω(∆(π, iσn), ∂ϕΓ −_{(π, iσ}

n))ei(−iσn−π)e−iπ =−Θ+₀.

Analogous considerations can be used to prove thatΘ−₀ = Θ+₀ when η <0.

Finally, let Γ˜± _∈ X1(Sh×D±_˜r) be two solutions of equation (3.4) asymptotic to Γˆ˜. Then it follows from Theorem 3.2.2 that there exist (ϕ0, τ0) ∈ C2 such that ˆ

˜

Γ(ϕ, τ) = ˆΓ(ϕ+ϕ0, τ+τ0). Thus, uniqueness of solutionsΓ˜±≍Γˆ˜ andΓ±≍Γˆ allows us to conclude thatΓ˜±(ϕ, τ) =Γ±(ϕ+ϕ0, τ +τ0). Therefore,

˜ Θ±₀ = lim Im(τ)→±∞Ω(˜Γ +_{(ϕ, τ)−Γ˜}−_{(ϕ, τ), ∂} ϕΓ˜−(ϕ, τ))e∓i(τ−ϕ) = lim Im(τ)→±∞Ω(∆(ϕ+ϕ0, τ+τ0), ∂ϕΓ −_(ϕ+ϕ 0, τ +τ0))e∓i(τ+τ0−(ϕ+ϕ0))e±i(τ0−ϕ0) = Θ±₀e±i(τ0−ϕ0)_.

Theorem 3.5.2 (Stokes constant). Let H0 be the space of analytic Hamiltonian func- tionsH:_{U →}C_{which have the same properties as described in the introduction of the}

present Chapter. For a givenH_∈H0the constantsΘ0±define a functionalK0:H0 →C according to the formula,

K0 =−Θ−0Θ+0.

In other words,K0 is independent of the choice of the parametrisations Γ±. Moreover,

K0 is independent of the coordinate system, i.e., if H˜ ∈ H0 is another Hamiltonian function which is conjugated toH, i.e., H˜ =H◦Ψ for some analytic symplectic map Ψ which fixes the origin Ψ(0) = 0 then _K0(H) = K0( ˜H). The number

p

K0(H) is known as the Stokes constant.

Proof. This Theorem follows directly from the previous Lemmas since all the freedom we have in the definition of the K0 comes from the freedom of the parametrisations

Γ±. As the parametrisations are defined up to translation in(ϕ, τ) we get the desired conclusion which follows from the third item of the previous Lemma. The coordinate independence also follows from similar considerations.

Remark 3.5.2.1. If H is real analytic then,

K0(H) =      Θ−₀2 if η >0, −Θ−₀2 if η <0.

In the stable case, i.e. η >0, the Stokes constant is equal toΘ−₀.

Remark 3.5.2.2. If the Stokes constant p_K0(H) does not vanish then the asymptotic formula (3.58) provides an exponentially small lower bound for the splitting distance

kΓ+(ϕ, τ)₋Γ−(ϕ, τ)_k. Thus implying that H is non-integrable and that the normal form transformation Φdiverges.

Corollary 3.5.2.1. If His real analytic and XH is reversible with respect to the involu- tion (3.16) then there exist parametrisationsΓ±:Sh×Dr±→C4 which are symmetric

in the sense thatΓ±_{(ϕ, τ) =S(Γ}±_{(−ϕ,¯ −τ¯))such that the corresponding constantΘ}−

0 is a purely imaginary number, i.e.,Re(Θ−₀) = 0.

Proof. It follows from Remark 3.2.2.2 and the reversibility of XH that there exists a formal solutionΓˆ _∈τ−1_T

C4[[τ−1]] of equation (3.4) such that, ˆ

Γ(ϕ, τ) =S(ˆΓ(−ϕ,¯ −τ¯)). (3.84) This formal solution is unique up to translation ϕ+π, that is, if Γˆ˜ is another formal solution of the same class satisfying (3.84) then there is a number k _{∈ {}0,1_} such that Γˆ˜(ϕ, τ) = ˆΓ(ϕ+kπ, τ). Now due to Theorem 3.4.1 and Theorem 3.4.2 there exist unique Γ± : Sh ×Dr± → C4 such that Γ± ≍ Γˆ. If we define Γ˜±(ϕ, τ) =

S(Γ±(−ϕ,¯ −τ¯)) and taking into account that H is real analytic we conclude that the functionsΓ˜±:Sh×Dr±→C4 are solutions of equation (3.4) and due to (3.84) we also have that Γ˜± ≍Γˆ. Thus, uniqueness of Γ± implies that S(Γ±(−ϕ,¯ −τ¯)) =Γ±(ϕ, τ) yielding the first part of the corollary. As for the second part, taking into account the previous Theorem, we can writeΘ−₀ as follows,

Θ−₀ = lim

n→+∞Θ(0,−iσn)e

σn_,

whereσn is any real sequence such thatσn→+∞ asn→+∞. Thus, Θ−₀ = lim n→+∞Ω(∆(0,−iσn), ∂τΓ −_(0,−iσ n))eσn = lim n→+∞Ω(S(∆(0,−iσn)),S(∂τΓ −_(0,−iσ n)))eσn =₋ lim n→+∞Ω(∆(0,−iσn), ∂τΓ −_(0, −iσn))eσn =₋Θ−₀.

Remark 3.5.2.3. In fact the parametrisationsΓ± of the previous Corollary are uniquely defined by the reversibility up to a translationϕ+π in the first argument.