Strong nonlinear instability and growth of Sobolev norms near quasiperiodic finite-gap tori for the 2D cubic NLS equation

We consider the defocusing cubic nonlinear Schr\"odinger equation (NLS) on the two-dimensional torus. The equation admits a special family of elliptic invariant quasiperiodic tori called finite-gap solutions. These are inherited from the integrable 1D model (cubic NLS on the circle) by considering solutions that depend only on one variable. We study the long-time stability of such invariant tori for the 2D NLS model and show that, under certain assumptions and over sufficiently long timescales, they exhibit a strong form of transverse instability in Sobolev spaces $H^s(\mathbb{T}^2)$ ($0<s<1$). More precisely, we construct solutions of the 2D cubic NLS that start arbitrarily close to such invariant tori in the $H^s$ topology and whose $H^s$ norm can grow by any given factor. This work is partly motivated by the problem of infinite energy cascade for 2D NLS, and seems to be the first instance where (unstable) long-time nonlinear dynamics near (linearly stable) quasiperiodic tori is studied and constructed.


Introduction
A widely held principle in dynamical systems theory is that invariant quasiperiodic tori play an important role in understanding the complicated long-time behavior of Hamiltonian ODE and PDE. In addition to being important in their own right, the hope is that such quasiperiodic tori can play an important role in understanding other, possibly more generic, dynamics of the system by acting as islands in whose vicinity orbits might spend long periods of time before moving to other such islands. The construction of such invariant sets for Hamiltonian PDE has witnessed an explosion of activity over the past thirty years after the success of extending KAM techniques to infinite dimensions. However, the dynamics near such tori is still poorly understood, and often restricted to the linear theory. The purpose of this work is to take a step in the direction of understanding and constructing non-trivial nonlinear dynamics in the vicinity of certain quasiperiodic solutions for the cubic defocusing NLS equation. In line with the above philosophy emphasizing the role of invariant quasiperiodic tori for other types of behavior, another aim is to push forward a program aimed at proving infinite Sobolev norm growth for the 2D cubic NLS equation, an outstanding open problem. where (x, y) ∈ T 2 = R 2 /(2πZ) 2 , t ∈ R and u : R × T 2 → C. All the results in this paper extend trivially to higher dimensions d ≥ 3 by considering solutions that only depend on two variables 1 . This is a Hamiltonian PDE with conserved quantities: i) the Hamiltonian |∇u(x, y)| 2 + 1 2 |u(x, y)| 4 dx dy, We shall exhibit solutions whose energy moves from very high frequencies towards low frequencies (backward or inverse cascade), as well as ones that exhibit cascade in the opposite direction (forward or direct cascade). Such cascade phenomena have attracted a lot of attention in the past few years as they are central aspects of various theories of turbulence for nonlinear systems. For dispersive PDE, this goes by the name of wave turbulence theory which predicts the existence of solutions (and statistical states) of (2D-NLS) that exhibit a cascade of energy between very different length-scales. In the mathematical community, Bourgain drew attention to such questions of energy cascade by first noting that it can be captured in a quantitative way by studying the behavior of the Sobolev norms of the solution In his list of Problems on Hamiltonian PDE [Bou00], Bourgain asked whether there exist solutions that exhibit a quantitative version of the forward energy cascade, namely solutions whose Sobolev norms H s , with s > 1, are unbounded in time We should point out here that such growth cannot happen for s = 0 or s = 1 due to the conservation laws of the equations. For other Sobolev indices, there exists polynomial upper bounds for the growth of Sobolev norms (cf. [Bou96,Sta97,CDKS01,Bou04,Zho08,CW10,Soh11a,Soh12,Soh11b,CKO12,PTV17]). Nevertheless, results proving actual growth of Sobolev norms are much more scarce. After seminal works by Bourgain himself [Bou96] and Kuksin [Kuk96,Kuk97a,Kuk97b], the landmark result in [CKS + 10] played a fundamental importance in the recent progress, including this work: It showed that for any s > 1, δ 1, K 1, there exist solutions u of (2D-NLS) such that (1.5) u(0) H s ≤ δ and u(T ) H s ≥ K for some T > 0. Even if not mentioned in that paper, the same techniques also lead to the same result for s ∈ (0, 1). This paper induced a lot of activity in the area [ The above-cited works revealed an intimate connection between Lypunov instability and Sobolev norm growth. Indeed, the solution u = 0 of (2D-NLS) is an elliptic critical point and is linearly stable in all H s . From this point of view, the result in [CKS + 10] given in (1.5) can be interpreted as the Lyapunov instability in H s , s = 1, of the elliptic critical point u = 0 (the first integrals (1.1) and (1.2) imply Lyapunov stability in the H 1 and L 2 topology). It turns out that this connection runs further, particularly in relation to the question of finding solutions exhibiting (1.4). As was observed in [Han14], one way to prove the existence of such solutions is to prove that, for sufficiently many φ ∈ H s , an instability similar to that in (1.5) holds, but with u(0) − φ H s ≤ δ. In other words, proving long-time instability as in (1.5) but with solutions starting δ−close to φ, and for sufficiently many φ ∈ H s implies the existence (and possible genericness) of unbounded orbits satisfying (1.4). Such a program (based on a Baire-Category argument) was applied successfully for the Szegö equation on T in [GG15].
Motivated by this, one is naturally led to studying the Lyapunov instability of more general invariant objects of (2D-NLS) (or other Hamiltonian PDEs), or equivalently to investigate whether one can achieve Sobolev norm explosion starting arbitrarily close to a given invariant object. The first work in this direction is by one of the authors [Han14]. He considers the plane waves u(t, x) = Ae i(mx−ωt) with ω = m 2 + A 2 , periodic orbits of (2D-NLS), and proves that there are orbits which start δ-close to them and undergo H s Sobolev norm explosion, 0 < s < 1. This implies that the plane waves are Lyapunov unstable in these topologies. Stability results for plane waves in H s , s > 1, on shorter time scales are provided in [FGL14].
The next step in this program would be to study such instability phenomena near higher dimensional invariant objects, namely quasiperiodic orbits. This is the purpose of this work, in which we will address this question for the family of finite-gap tori of (1D-NLS) as solutions to the (2D-NLS). To control the linearized dynamics around such tori, we will impose some Diophantine (strongly non-resonant) conditions on the quasiperiodic frequency parameters. This allows us to obtain a stable linearized operator (at least with respect to the perturbations that we consider), which is crucial to control the delicate construction of the unstable nonlinear dynamics.
1.3. Statement of results. Roughly speaking, we will construct solutions to (2D-NLS) that start very close to the finite-gap tori in appropriate topologies, and exhibit either backward cascade of energy from high to low frequencies, or forward cascade of energy from low to high frequencies. In the former case, the solutions that exhibit backward cascade start in an arbitrarily small vicinity of a finite-gap torus in Sobolev spaces H s (T 2 ) with 0 < s < 1, but grow to become larger than any pre-assigned factor K 1 in the same H s (higher Sobolev norms H s with s > 1 decrease, but they are large for all times). In the latter case, the solutions that exhibit forward cascade start in an arbitrarily small vicinity of a finite-gap torus in L 2 (T 2 ), but their H s Sobolev norm (for s > 1) exhibits a growth by a large multiplicative factor K 1 after a large time. We shall comment further on those results after we state the theorems precisely.
To do that, we need to introduce the Birkhoff coordinates for equation 1D-NLS. Grébert and Kappeler showed in [GK14a] that there exists a globally defined map, called the Birkhoff map, such that ∀s ≥ 0 Therefore in these coordinates, called Birkhoff coordinates, equation (1D-NLS) becomes a chain of nonlinear harmonic oscillators and it is clear that the phase space is foliated by finite and infinite dimensional tori with periodic, quasiperiodic or almost periodic dynamics, depending on how many of the actions I m (which are constant!) are nonzero and on the properties of rational dependence of the frequencies.
In this paper we are interested in the finite dimensional tori with quasiperiodic dynamics. Fix d ∈ N and consider a set of modes Fix also a value for the actions . . , d, z m = 0 for m ∈ S 0 , which is supported on the set S 0 . Any orbit on this torus is quasiperiodic (or periodic if the frequencies of the rigid rotation are completely resonant). We will impose conditions to have non-resonant quasiperiodic dynamics. This will imply that the orbits on T d are dense. By equation (1.7), it is clear that this torus, as an invariant object of equation 1D-NLS, is stable for this equation for all times in the sense of Lyapunov.
The torus (1.9) (actually, its pre-image Φ −1 (T d ) though the Birkhoff map) is also an invariant object for the original equation (2D-NLS). The main result of this paper will show the instability (in the sense of Lyapunov) of this invariant object. Roughly speaking, we show that under certain assumptions (on the choices of modes (1.8) and actions (1.9)) these tori are unstable in the H s (T 2 ) topology for s ∈ (0, 1). Even more, there exist orbits which start arbitrarily close to these tori and undergo an arbitrarily large H s -norm explosion.
We will abuse notation, and identify H s (T) with the closed subspace of H s (T 2 ) of functions depending only on the x variable. Consequently, Theorem 1.1. Fix a positive integer d. For any choice of d modes S 0 (see (1.8)) satisfying a genericity condition (namely Definition 4.1 with sufficiently large L), there exists ε * > 0 such that for any ε ∈ (0, ε * ) there exists a positive measure Cantor-like set I ⊂ (ε/2, ε) d of actions, for which the following holds true for any torus T d = T d (S 0 , I 0 m ) with I 0 m ∈ I: (1) For any s ∈ (0, 1), δ > 0 small enough, and K > 0 large enough, there exists an orbit u(t) of (2D-NLS) and a time (2) For any s > 1, and any K > 0 large enough, there exists an orbit u(t) of (2D-NLS) and a time Here σ, σ > 0 are independent of K.
1.4. Comments and remarks on Theorem 1.1: (1) The relative measure of the set I of admissible actions can be taken as close to 1 as desired. Indeed, by taking smaller ε * , one has that the relative measure satisfies for some constant C > 0 and 0 < κ < 1 independent of ε * > 0. The genericity condition on the set S 0 and the actions (I m ) m∈S 0 ∈ I ensure that the linearized dynamics around the resulting torus T d is stable for the perturbations we need to induce the nonlinear instability. In fact, a subset of those tori is even linearly stable for much more general perturbations as we remark below.
(2) Why does the finite gap solution need to be small? To prove Theorem 1.1 we need to analyze the linearization of equation (2D-NLS) at the finite gap solution (see Section 4). Roughly speaking, this leads to a Schrödinger equation with a quasi-periodic potential. Luckily, such operators can be reduced to constant coefficients via a KAM scheme. This is known as reducibility theory which allows one to construct a change of variables that casts the linearized operator into an essentially constant coefficient diagonal one. This KAM scheme was carried out in [MP18], and requires the quasi-periodic potential, given by the finite gap solution here, to be small for the KAM iteration to converge. That being said, we suspect a similar result to be true for non-small finite gap solutions.
(3) To put the complexity of this result in perspective, it is instructive to compare it with the stability result in [MP18]. In that paper, it is shown that a proper subset I ⊂ I of the tori considered in Theorem 1.1 are Lyapunov stable in H s , s > 1, but for shorter time scales than those considered in this theorem. More precisely, all orbits that are initially δ-close to T d in H s stay Cδ-close for some fixed C > 0 for time scales t ∼ δ −2 . The same stability result (with a completely identical proof) holds if we replace H s by F 1 norm (functions whose Fourier series is in 1 ). In fact, by trivially modifying the proof, one could also prove stability on the δ −2 timescale in F 1 ∩ H s for 0 < s < 1. What this means is that the solutions in the first part of Theorem 1.1 remains within Cδ of T d up to times ∼ δ −2 but can diverge vigorously afterwards at much longer time scales. It is also worth mentioning that the complementary subset I \ I has a positive measure subset where tori are linearly unstable since they possess a finite set of modes that exhibit hyperbolic behavior. In principle, hyperbolic directions are good for instability, but they are not useful for our purposes since they live at very low frequencies, and hence cannot be used (at least not by themselves alone) to produce a substantial growth of Sobolev norms. We avoid dealing with these linearly unstable directions by restricting our solution to an invariant subspace on which these modes are at rest. (4) It is expected that a similar statement to the first part of Theorem 1.1 is also true for s > 1.
This would be a stronger instability compared to that in the second part (for which the initial perturbation is small in L 2 but not in H s ). Nevertheless, this case cannot be tackled with the techniques considered in this paper. Indeed, one of the key points in the proof is to perform a (partial) Birkhoff normal form up to order 4 around the finite gap solution. The terms which lead to the instabilities in Theorem 1.1 are quasi-resonant instead of being completely resonant. Working in the H s topology with s ∈ (0, 1), such terms can be considered completely resonant with little error on the timescales where instability happens. However, this cannot be done for s > 1, for which one might be able to eliminate those terms by a higher order normal form (s > 1 gives a stronger topology and can thus handle worse small divisors). This would mean that one needs other resonant terms to achieve growth of Sobolev norms. The same difficulties were encountered in [Han14] to prove the instability of the plane waves of (2D-NLS). (5) For finite dimensional Hamiltonian dynamical systems, proving Lyapunov instability for quasiperiodic Diophantine elliptic (or maximal dimensional Lagrangian) tori is an extremely difficult task. Actually all the obtained results [CZ13,GK14b] deal with C r or C ∞ Hamiltonians, and not a single example of such instability is known for analytic Hamiltonian systems. In fact, there are no results of instabilities in the vicinity of non-resonant elliptic critical points or periodic orbits for analytic Hamiltonian systems (see [LCD83,Dou88,KMV04] for results on the C ∞ topology). The present paper proves the existence of unstable Diophantine elliptic tori in an analytic infinite dimensional Hamiltonian system. Obtaining such instabilities in infinite dimensions is, in some sense, easier: having infinite dimensions gives "more room" for instabilities. (6) It is well known that many Hamiltonian PDEs possess quasiperiodic invariant tori [Way90, Most of these tori are normally elliptic and thus linearly stable. It is widely expected that the behavior given by Theorem 1.1 also arises in the neighborhoods of (many of) those tori. Nevertheless, it is not clear how to apply the techniques of the present paper to these settings.
1.5. Scheme of the proof. Let us explain the main steps to prove Theorem 1.1.
(1) Analysis of the 1-dimensional cubic Schrödinger equation. We express the 1-dimensional cubic NLS in terms of the Birkhoff coordinates. We need a quite precise knowledge of the Birkhoff map (see Theorem 3.1). In particular, we need that it "behaves well" in 1 . This is done in the paper [Mas18b] and summarized in Section 3. In Birkhoff coordinates, the finite gap solutions are supported in a finite set of variables. We use such coordinates to express the Hamiltonian (1.1) in a more convenient way.
(2) Reducibility of the 2-dimensional cubic NLS around a finite gap solution. We reduce the linearization of the vector field around the finite gap solutions to a constant coefficients diagonal vector field. This is done in [MP18] and explained in Section 4. In Theorem 4.4 we give the conditions to achieve full reducibility. In effect, this transforms the linearized operator around the finite gap into a constant coefficient diagonal (in Fourier space) operator, with eigenvalues {Ω  } ∈Z 2 \S 0 . We give the asymptotics of these eigenvalues in Theorem 4.6, which roughly speaking look like (1.10) for frequencies  = (m, n) satisfying |m|, |n| ∼ J. This seemingly harmless O(J −2 ) correction to the unperturbed Laplacian eigenvalues is sharp and will be responsible for the restriction to s ∈ (0, 1) in the first part of Theorem 1.1 as we shall explain below. (3) Degree three Birkhoff normal form around the finite gap solution. This is done in [MP18], but we shall need more precise information from this normal form that will be crucial for Steps 5 and 6 below. This is done in 5 (see Theorem 5.2). (4) Partial normal form of degree four. We remove all degree four monomials which are not (too close to) resonant. This is done in Section 6, and leaves us with a Hamiltonian with (close to) resonant degree-four terms plus a higher-degree part which will be treated as a remainder in our construction. (5) We follow the paradigm set forth in [CKS + 10, GK15] to construct solutions to the truncated Hamiltonian consisting of the (close to) resonant degree-four terms isolated above, and then afterwards to the full Hamiltonian by an approximation argument. This construction will be done at frequencies  = (m, n) such that |m|, |n| ∼ J with J very large, and for which the dynamics is effectively given by the following system of ODE We remark that the conditions of the set R( ) are essentially equivalent to saying that (  1 ,  2 ,  3 , ) form a rectangle in Z 2 . Also note that by the asymptotics of Ω  mentioned above in (1.10), one obtains that Γ = O(J −2 ) if all the frequencies involved are in R( ) and satisfy |m|, |n| ∼ J.
The idea now is to reduce this system into a finite dimensional system called the "Toy Model" which is tractable enough for us to construct a solution that cascades energy. An obstruction to this plan is presented by the presence of the oscillating factor e iΓt for which Γ is not zero (in contrast to [CKS + 10]) but rather O(J −2 ). The only way to proceed with this reduction is to approximate e iΓt ∼ 1 which is only possible provided J −2 T 1. The solution coming from the Toy Model is supported on a finite number of modes  ∈ Z 2 \ S 0 satisfying |j| ∼ J, and the time it takes for the energy to diffuse across its modes is T ∼ O(ν −2 ) where ν is the characteristic size of the modes in 1 norm. Requiring the solution to be initially close in H s to the finite gap would necessitate that νJ s δ which gives that T δ J −2s , and hence the condition J −2 T 1 translates into the condition s < 1. This explains the restriction to s < 1 in the first part of Theorem 1.1. If we only require our solutions to be close to the finite gap in L 2 , then no such restriction on ν is needed, and hence there is no restriction on s beyond being s > 0 and s = 1, which is the second part of the theorem. This analysis is done in Section 7 and 8. In the former, we perform the reduction to the effective degree 4 Hamiltonian taking into account all the changes of variables performed in the previous sections; while in Section 8 we perform the above approximation argument allowing to shadow the Toy Model solution mentioned above with a solution of (2D-NLS) exhibiting the needed norm growth, thus completing the proof of Theorem 1.1. 2. Notation and functional setting 2.1. Notation. For a complex number z, it is often convenient to use the notation For any subset Γ ⊂ Z 2 , we denote by h s (Γ) the set of sequences (a  ) ∈Γ with norm Our phase space will be obtained by an appropriate linearization around the finite gap solution with d frequencies/actions. For a finite set S 0 ⊂ Z × {0} of d elements, we consider the phase space X = (C d × T d ) × 1 (Z 2 \ S 0 ) × 1 (Z 2 \ S 0 ). The first part (C d × T d ) corresponds to the finite-gap sites in action angle coordinates, whereas 1 (Z 2 \ S 0 ) × 1 (Z 2 \ S 0 ) corresponds to the remaining orthogonal sites in frequency space. We shall often denote the 1 norm by · 1 . We shall denote variables on X by We shall use multi-index notation to write monomials like Y l and m α,β = a αāβ where l ∈ N d and α, β ∈ (N) Z 2 \S 0 . Often times, we will abuse notation, and simply use the notation a ∈ 1 to mean a = (a,ā) ∈ 1 (Z 2 \ S 0 ) × 1 (Z 2 \ S 0 ), and a 1 = a 1 (Z 2 \S 0 ) .
Definition 2.1. For a monomial of the form e i ·θ Y l m α,β , we define its degree to be 2|l| + |α| + |β| − 2, where the modulus of a multi-index is given by its 1 norm.

Regular Hamiltonians.
Given a Hamiltonian function F (Y, θ, a) on the phase space X , we associate to it the Hamiltonian vector field where we have used the standard complex notation to denote the Fréchet derivatives of F with respect to the variable a ∈ 1 .
We will often need to complexify the variable θ ∈ T d into the domain T d ρ := {θ ∈ C d : Re(θ) ∈ T d , |Im(θ)| ≤ ρ} and consider vector fields which are functions from , X (ā) ) which are analytic in Y, θ, a. Our vector fields will be defined on the domain On the vector field, we use as norm All Hamiltonians F considered in this article are analytic, real valued and can be expanded in Taylor Fourier series which are well defined and pointwise absolutely convergent Correspondingly we expand vector fields in Taylor Fourier series (again well defined and pointwise absolutely convergent) To a vector field we associate its majorant α,β, | e ρ | | Y l m α,β and require that this is an analytic map on D(r). Such a vector field is called majorant analytic. Since Hamiltonian functions are defined modulo constants, we give the following definition of the norm of F : Note that the norm | · | ρ,r controls the | · | ρ ,r whenever ρ < ρ, r < r.
Finally, we will also consider Hamiltonians F (λ; θ, a,ā) ≡ F (λ) depending on an external parameter λ ∈ O ⊂ R d . For those, we define the inhomogeneous Lipschitz norm: 2.3. Commutation rules. Given two Hamiltonians F and G, we define their Poisson bracket as {F, G} := dF (X G ); in coordinates Given α, β ∈ N Z 2 \S 0 we denote m α,β := a αāβ . To the monomial e i ·θ Y l m α,β with ∈ Z d , l ∈ N d we associate various numbers. We denote by We also associate to e i ·θ Y l m α,β the quantities π(α, β) = (π x , π y ) and π( ) defined by The above quantities are associated with the following mass M and momentum P = (P x , P y ) functionals given by Remark 2.2. An analytic hamiltonian function F (expanded as in (2.2)) commutes with the mass M and the momentum P if and only if the following selection rules on its coefficients hold: where η(α, β), η( ) are defined in (2.3) and π(α, β), π( ) are defined in (2.4).
Definition 2.3. We will denote by A ρ,r the set of all real-valued Hamiltonians of the form (2.2) with finite | · | ρ,r norm and which Poisson commute with M, P. Given a compact set O ⊂ R d , we denote by A O ρ,r the Banach space of Lipschitz maps O → A ρ,r with the norm | · | O ρ,r . From now on, all our Hamiltonians will belong to some set A ρ,r for some ρ, r > 0.

Adapted variables and Hamiltonian formulation
3.1. Fourier expansion and phase shift. Let us start by expanding u in Fourier coefficients Then, the Hamiltonian H 0 introduced in (1.1) can be written as where the means the sum over the quadruples is a constant of motion, we make a trivial phase shift and consider an equivalent Hamiltonian corresponding to the Hamilton equation Clearly the solutions of (3.2) differ from the solutions of (2D-NLS) only by a phase shift 3 . Then, 3.2. The Birkhoff map for the 1D cubic NLS. We devote this section to gathering some properties of the Birkhoff map for the integrable 1D NLS equation. These will be used to write the Hamiltonian (3.3) in a more convenient way. The main reference for this section is [Mas18b]. We shall denote by B s (r) the ball of radius r and center 0 in the topology of h s ≡ h s (Z).
Theorem 3.1. There exist r * > 0 and a symplectic, real analytic map Φ with dΦ(0) = I such that ∀s ≥ 0 one has the following The same estimate holds for Φ −1 − I or by replacing the space h s with the space 1 .
(ii) Moreover, if q ∈ h s for s ≥ 1, Φ introduces local Birkhoff coordinates for (NLS-1d) in h s as follows: the integrals of motion of (NLS-1d) are real analytic functions of the actions In order to show the equivalence we consider any solution u(x, t) of (3.2) and consider the invertible map Then a direct computation shows that v solves 2D-NLS.
have the form ∂Im , ∀m ∈ Z. Then one has the asymptotic expansion where m (I) is at least quadratic in I.
Proof. Item (i) is the main content of [Mas18b], where it is proved that the Birkhoff map is majorant analytic between some Fourier-Lebesgue spaces. Item (ii) is proved in [GK14a]. Item (iii) is Theorem 1.3 of [KST17].
To begin with, we start from the Hamiltonian in Fourier coordinates (3.3), and set We rewrite the Hamiltonian accordingly in increasing degree in a, obtaining Step 1: First we do the following change of coordinates, which amounts to introducing Birkhoff coordinates on the line Z × {0}. We set In those new coordinates, the Hamiltonian becomes where Step 2: Next, we go to action-angle coordinates only on the set In those coordinates, the Hamiltonian becomes (using (3.4)) Step 3: Now, we expand each line by itself. By Taylor expanding around the finite-gap torus corresponding to (Y, θ, a) = (0, θ, 0) we obtain, up to an additive constant, where we have used formula (3.4) in order to deduce that ∂ 2 h nls1d ∂Im∂In (0) = −δ m n where δ m n is the Kronecker delta.

Lemma 3.3 (Frequencies around the finite gap torus).
Denote Then, (1) The map (I 1 , . . . , I d ) → λ(I 1 , . . . , I d ) = ( λ i (I 1 , . . . , I d )) 1≤i≤d is a diffeomorphism from a small neighborhood of 0 of R d to a small neighborhood of 0 in R d . Indeed, λ =Identity +(quadratic in I). More precisely, there exists ε 1d > 0 such that if 0 < ε < ε 1d and . From now on, and to simplify notation, we will use the vector λ as a parameter as opposed to (I 1 , . . . , I d ), and we shall set the vector to denote the frequencies at the tangential sites in S 0 .
Proceeding as in [MP18], one can prove the following result:

Reducibility theory of the quadratic part
In this section, we review the reducibility of the quadratic part N + H (0) (see (3.14) and (3.15)) of the Hamiltonian, which is the main part of the work [MP18]. This will be a symplectic linear change of coordinates that transforms the quadratic part into an effectively diagonal, time independent expression. 4.1. Restriction to an invariant sublattice Z 2 N . For N ∈ N, we define the sublattice Z 2 N := Z×N Z and remark that it is invariant for the flow in the sense that the subspace is invariant for the original NLS dynamics and that of the Hamiltonian (3.13). From now on, we restrict our system to this invariant sublattice, with The reason for this restriction is that it simplifies (actually eliminates the need for) some genericity requirements that are needed for the work [MP18] as well as some of the normal forms that we will perform later. It will also be important to introduce the following two subsets of Z 2 N : Definition 4.1 (L−genericity). Given L ∈ N, we say that S 0 is L-generic if it satisfies the condition

4.2.
Admissible monomials and reducibility. The reducibility of the quadratic part of the Hamiltonian will introduce a change of variables that modifies the expression of the mass M and momentum P as follows. Let us set (4.4) These will be the expressions for the mass and momentum after the change of variables introduced in the following two theorems. Notice the absence of the terms 1≤i≤d from the expressions of M and P x above. These terms are absorbed in the new definition of the Y and a variables.
(2) For each λ ∈ C (0) and all r ∈ [0, r 0 ], ρ ∈ [ ρ 0 64 , ρ 0 ], there exists an invertible symplectic change of variables L (0) , that is well defined and majorant analytic from D(ρ/8, ζ 0 r) → D(ρ, r) (here ζ 0 > 0 is a constant depending only on ρ 0 , max(|m k | 2 )) and such that if a ∈ h 1 (Z 2 N \ S 0 ), then (3) The mass M and the momentum P (defined in (2.5)) in the new coordinates are given by (4) The map L (0) maps h 1 to itself and has the following form The same holds for the inverse map (L (0) ) −1 . (5) The linear maps L(λ; θ, ε) and Q(λ; θ, ε) are block diagonal in the y Fourier modes, in the sense that L = diag n∈N N (L n ) with each L n acting on the sequence {a (m,n) , a (m,−n) } m∈Z (and similarly for Q). Moreover, L 0 = Id and L n is of the form Id + S n where S n is a smoothing operator in the following sense: with the smoothing norm · ρ,−1 defined in (4.8) below where P {|m|≥K} is the orthogonal projection of a sequence (c m ) m∈Z onto the modes |m| ≥ K.
The above smoothing norm is defined as follows: Let S(λ; θ, ε) be an operator acting on sequences (c k ) k∈Z through its matrix elements S(λ; θ, ε) m,k . Let us denote by S(λ; , ε) m,k the θ-Fourier coefficients of S(λ; θ, ε) m,k . For ρ, ν > 0 we define S(λ; θ, ε) ρ,ν as: This definition is equivalent to the more general norm used in Definition 3.9 of [MP18]. Roughly speaking, the boundedness of this norm means that, in terms of its action on sequences, S maps k ν 1 → 1 . As observed in Remark 3.10 of [MP18], thanks to the conservation of momentum this also means that S maps 1 → k −ν 1 .
Remark 4.5. Note that in [MP18] Theorem 4.4 is proved in h s norm with s > 1, for instance in (4.8) the 1 norm is substituted with the h s one. However the proof only relies on momentum conservation and on the fact that h s is an algebra w.r.t. convolution, which holds true also for 1 . Hence the proof of our case is identical and we do not repeat it.
We are able to describe quite precisely the asymptotics of the frequencies Ω  of Theorem 4.4.
Here the {µ i (λ)} 1≤i≤d are the roots of the polynomial Finally Theorems 4.4 and 4.6 follow from Theorems 5.1 and 5.3 of [MP18], together with the observation that the set C defined in Definition 2.3 of [MP18] We conclude this section with a series of remarks.
Remark 4.7. Notice that the {µ i (λ)} 1≤i≤d depend on the number d of tangential sites but not on the {m i } 1≤i≤d .
Remark 4.8. The asymptotic expansion (4.9) of the normal frequencies does not contain any constant term. The reason is that we canceled such a term when we subtracted the quantity M (u) 2 from the Hamiltonian at the very beginning (see the footnote in Section 3.1). Of course if we had not removed M (u) 2 , we would have had a constant correction to the frequencies, equal to q(ωt, ·) 2 L 2 . Since q(ωt, x) is a solution of (2D-NLS), it enjoys mass conservation, and thus q(ωt, ·) 2 L 2 = q(0, ·) 2 L 2 is independent of time.
Remark 4.9. In the new variables, the selection rules of Remark 2.2 become (with H expanded as in (2.2)):

Elimination of cubic terms
If we apply the change L (0) obtained in Theorem 4.4 to Hamiltonian (3.13), we obtain As a direct consequence of Lemma 3.4 and Theorem 4.4, estimates (3.19) hold also for K (j) , j = 1, 2 and K (≥3) . We now perform one step of Birkhoff normal form change of variables which cancels out K (1) completely. In order to define such a change of variables we need to impose third order Melnikov conditions, which hold true on a subset of the set C (0) of Theorem 4.4.
This lemma is proven in Appendix C of [MP18]. The main result of this section is the following theorem.
Lemma 5.3. For every ρ, r > 0 the following holds true: (i) Let h, f ∈ A O ρ,r . For any 0 < ρ < ρ and 0 < r < r, one has where υ := min 1 − r r , ρ − ρ . If υ −1 |f | O ρ,r < ζ sufficiently small then the (time-1 flow of the) Hamiltonian vector field X f defines a close to identity canonical change of variables T f such that |h for all 0 < ρ < ρ , 0 < r < r .
(ii) Let f, g ∈ A O ρ,r of minimal degree respectively d f and d g (see Definition 2.1) and define the function Then T i (f ; g) is of minimal degree d f i + d g and we have the bound Proof of Theorem 5.2. We look for L (1) as the time-one-flow of a Hamiltonian χ (1) . With N := ω · Y + ∈Z 2 We choose χ (1) to solve the homological equation { N , χ (1) } + K (1) = 0. Thus we set .
since the terms q fg m appearing in H (1) (and hence K (1) ) are O( √ ε). We come to the terms of line (5.8). First we use the homological equation { N , χ (1) } + K (1) = 0 to get that Therefore, we set Q (2) as in (5.3) and By Lemma 5.3, Q (≥3) has degree at least 3 and fulfills the quantitative estimate (5.4). To prove (iv), we use the fact that { M, χ (1) } = { P, χ (1) } = 0 follows since K (1) commutes with M and P, hence its monomials fulfill the selection rules of Remark 4.9. By the explicit formula for χ (1) above, it follows that the same selection rules hold for χ (1) , and consequently L (1) preserves M and P.

Analysis of the quartic part of the Hamiltonian
At this stage, we are left with the Hamiltonian Q given in (5.2). The aim of this section is to eliminate non-resonant terms from Q (2) . First note that Q (2) contains monomials which have one of the two following forms with |l| = 1.
In order to cancel out the terms quadratic in a by a Birkhoff Normal form procedure, we only need the second Melnikov conditions imposed in (4.6). In order to cancel out the quartic tems in a we need fourth Melnikov conditions, namely to control expressions of the form We start by defining the following set R 4 ⊂ A 4 (see Definition 4.2), R 4 := (j, , σ) : = 0 and  1 ,  2 ,  3 ,  4 / ∈ S form a rectangle (6.2) = 0 and  1 ,  2 / ∈ S ,  3 ,  4 ∈ S form a horizontal rectangle (even degenerate) = 0,  1 ,  2 ,  3 ∈ S ,  4 ∈ S and |m 4 | < M 0 , where M 0 is a universal constant = 0,  1 ,  2 ,  3 ,  4 ∈ S form a horizontal trapezoid where S is the set defined in (4.2). Here a trapezoid (or a rectangle) is said to be horizontal if two sides are parallel to the x-axis.
Proposition 6.1. Fix 0 < ε 2 < ε 1 sufficiently small and τ 2 > τ 1 sufficiently large. There exist positive γ 2 > 0, L 2 ≥ L 1 (with L 2 depending only on d), such that for all 0 < ε ≤ ε 2 and for an L 2 -generic choice of the set S 0 (in the sense of Definition 4.1), the set has positive measure and C (1) \ C (2) ε κ 2 2 for some κ 2 > 0 independent of ε 2 . The proof of the proposition, being quite technical, is postponed to Appendix A. An immediate consequence, following the same strategy as for the proof of Theorem 5.2, is the following result. We define Π R 4 as the projection of a function in D(ρ, r) onto the sum of monomials with indexes in R 4 . Abusing notation, we define analogously Π R 2 as the projection onto monomials e i ·θ Y l a σ 1  1 a σ 2  2 with |l| = 1 and (  1 ,  2 , , σ 1 , σ 2 ) ∈ R 2 . Figure 1. The black dots, are the points in S 0 . The two rectangles and the trapezoid correspond to cases 1,2,4 in R 4 . In order to represent case 3. we have highlighted three points in S. To each such triple we may associate at most one = 0 and one  4 ∈ Z, which form a resonance of type 3.

Construction of the toy model
Once we have performed (partial) Birkhoff normal form up to order 4, we can start applying the ideas developed in [CKS + 10] to Hamiltonian (6.3). Note that throughout this section ε > 0 is a fixed parameter. Namely, we do not use its smallness and we do not modify it.
We first perform the (time dependent) change of variables to rotating coordinates to the Hamiltonian (6.3), which leads to the corrected Hamiltonian We split this Hamiltonian as a suitable first order truncation G plus two remainders, where Q (2) Res and Q (≥3) are the Hamiltonians introduced in Theorem 6.2. For the rest of this section we focus our study on the truncated Hamiltonian G. Note that the remainder J 1 is not smaller than G. Nevertheless it will be smaller when evaluated on the particular solutions we consider. The term R is smaller than G for small data since it is the remainder of the normal form obtained in Theorem 6.2. Later in Section 8 we show that including the dismissed terms J 1 and R barely alters the dynamics of the solutions of G that we analyze. 7.1. The finite set Λ. We now start constructing special dynamics for the Hamiltonian G with the aim of treating the contributions of J 1 and R as remainder terms. Following [CKS + 10], we do not study the full dynamics of G but we restrict the dynamics to invariant subspaces. Indeed, we shall construct a set Λ ⊂ Z := (Z × N Z) \ (S 0 ∪ S ) for some large N , in such a way that it generates an invariant subspace (for the dynamics of G) given by Thus, we consider the following definition.
Definition 7.1 (Completeness). We say that a set Λ ⊂ Z is complete if U Λ is invariant under the dynamics of G.
Remark 7.2. It can be easily seen that if Λ is complete, U Λ is also invariant under the dynamics of G + J 1 .
We construct a complete set Λ ⊂ Z (see Definition 7.1) and we study the restriction on it of the dynamics of the Hamiltonian G in (7.3). Following [CKS + 10], we impose several conditions on Λ to obtain dynamics as simple as possible.
The set Λ is constructed in two steps. First we construct a preliminary set Λ 0 ⊂ Z 2 on which we impose numerous geometrical conditions. Later on we scale Λ 0 by a factor N to obtain Λ ⊂ The set Λ 0 is "essentially" the one described in [CKS + 10]. The crucial point in that paper is to choose carefully the modes so that each mode in Λ 0 only belongs to two rectangles with vertices in Λ 0 . This allows to simplify considerably the dynamics and makes it easier to analyze. Certainly, this requires imposing several conditions on Λ 0 . We add some extra conditions to adapt the set Λ 0 to the particular setting of the present paper.
• Property V Λ 0 (Faithfulness): Apart from nuclear families, Λ 0 contains no other rectangles. In fact, by the closure property I Λ 0 , this also means that it contains no right angled triangles other than those coming from vertices of nuclear families. • Property VI Λ 0 : There are no two elements in Λ 0 such that  1 ±  2 = 0. There are no three elements in Λ 0 such that  1 −  2 +  3 = 0. If four points in Λ 0 satisfy  1 −  2 +  3 −  4 = 0 then either the relation is trivial or such points form a family. • Property VII Λ 0 : There are no points in Λ 0 with one of the coordinates equal to zero i.e.
• Property VIII Λ 0 : There are no two points in Λ 0 which form a right angle with 0. Condition I Λ 0 is just a rephrasing of the completeness condition introduced in Definition 7.1. Properties II Λ 0 , III Λ 0 , IV Λ 0 , V Λ 0 correspond to being a family tree as stated in [CKS + 10].
The construction of such kind of sets was done first in [CKS + 10] (see also [GK15,GK17,Gua14,GHP16]) where the authors construct sets Λ satisfying Properties I Λ -V Λ and estimate (7.8). The proof of Theorem 7.3 follows the same lines as the ones in those papers. Indeed, Properties VI Λ -VIII Λ can be obtained through the same density argument. Finally, the estimate (7.7), even if it is not stated explicitly in [CKS + 10], it is an easy consequence of the proof in that paper (in [GK15, GK17, GHP16] a slightly weaker estimate is used).
Remark 7.4. Note that s ∈ (0, 1) implies that were are constructing a backward cascade orbit (energy is transferred from high to low modes). This means that the modes in each generation of Λ 0 are just switched oppositely Λ 0j ↔ Λ 0,g−j+1 compared to the ones constructed in [CKS + 10]. The second statement of Theorem 1.1 considers s > 1 and therefore a forward cascade orbit (energy transferred from low to high modes). For this result, we need a set Λ 0 of the same kind as that of [CKS + 10], which thus satisfies instead of estimate (7.5).
We now scale Λ 0 by a factor N satisfying (4.1) and we denote by Λ := N Λ 0 . Note that the listed properties I Λ 0 -VIII Λ 0 are invariant under scaling. Thus, if they are satisfied by Λ 0 , they are satisfied by Λ too.
Lemma 7.5. There exists a set Λ satisfying all statements of Theorem 7.3 (with a different f (g) satisfying (7.6)) and also the following additional properties.
First we note that m 1 m 2 + n 1 n 2 = 0 by property VIII Λ 0 , since m = 0 cannot be a solution. Now consider the discriminant ∆ = (m 1 + m 2 ) 2 − 4(m 1 m 2 + n 1 n 2 ). If ∆ < 0, then no right angle is possible. If ∆ = 0, then clearly |m| ≥ 1/2, since once again m = 0 is not a solution. Finally let ∆ > 0. Then Denoting by γ := 4(m 1 m 2 +n 1 n 2 ) (m 1 +m 2 ) 2 , the condition ∆ > 0 implies that −∞ < γ < 1. Splitting in two cases: |γ| ≤ 1 and γ < −1 one can easily obtain that either way m satisfies (7.9). Now it only remains to scale the set Λ by a factor (f (g)) 4 . Then, taking as new f (g), f (g) := (f (g)) 5 , the obtained set Λ satisfies all statements of Theorem 7.3 and also the statements of Lemma 7.5. 7.2. The truncated Hamiltonian on the finite set Λ and the [CKS + 10] toy model. We use the properties of the set Λ given by Theorem 7.3 and Lemma 7.5 to compute the restriction of the Hamiltonian G in (7.3) to the invariant subset U Λ (see (7.4)).
Lemma 7.6. Consider the set Λ ⊂ N Z × N Z obtained in Theorem 7.3. Then, the set is invariant under the flow associated to the Hamiltonian G. Moreover, G restricted to M Λ can be written as and the remainder J 2 satisfies (7.12) |J 2 | ρ,r r 2 (f (g)) − 4 5 .
Proof. First we note that, since Y = 0 on M Λ , Res is the Hamiltonian defined in Theorem 6.2. We start by analyzing the Hamiltonian Q (2) introduced in Theorem 5.2, which is defined as We analyze each term. Here it plays a crucial role that Λ ⊂ N Z × N Z with N = f (g) 4/5 . In order to estimate K (2) , defined in (5.1), we recall that Λ does not have any mode in the x-axis and therefore the original quartic Hamiltonian has not been modified by the Birkhoff map (1.6) (this is evident from the formula for H (2) in (3.17)). Thus, it is enough to analyze how the quartic Hamiltonian has been modified by the linear change L (0) analyzed in Theorems 4.4 and 4.6. Using the smoothing property of the change of coordinates L (0) given in Statement 5 of Theorem 4.4, one obtains Now we deal with the term {K (1) , χ (1) }. Since we only need to analyze Π R 4 {K (1) , χ (1) } M Λ , we only need to consider monomials in K (1) and in χ (1) which have at least two indexes in Λ. We represent this by setting #Λ≥2 , where #Λ ≥ 2 means that we restrict to those monomials which have at least two indexes in Λ. We then have We estimate the size of χ #Λ≥2 has coefficients (7.13) χ (1) We first estimate the tails (in ) of χ (1) and then we analyze the finite number of cases left. For the tails, it is enough to use Theorem 5.2 to deduce the following estimate for any ρ ≤ ρ 1 /2, where ρ 1 is the constant introduced in that theorem, We restrict our attention to monomials with | | ≤ 4 √ N . We take  2 ,  3 ∈ Λ and we consider different cases depending on  1 and the properties of the monomial. In each case we show that the denominator of (7.13) is larger than N . Case 1. Suppose that  1 / ∈ S . The selection rules are (according to Remark 4.9) η( ) + σ 1 + σ 2 + σ 3 = 0 , m · + σ 1 m 1 + σ 2 m 2 + σ 3 m 3 = 0 , σ 1 n 1 + σ 2 n 2 + σ 3 n 3 = 0 and the leading term in the denominator of (7.13) is where m 2 = (m 2 1 , . . . , m 2 d ). We consider the following subcases: A1 σ 3 = σ 1 = +1, σ 2 = −1. In this case  1 −  2 +  3 − v = 0, where v := (− m · , 0). We rewrite (7.14) as Assume first  2 =  3 . Since the set Λ satisfies Lemma 7.5 1. and | m · | 4 √ N f (g) 1/5 , we can ensure that  2 and  3 do not form a right angle with v, thus Actually by the second statement of Lemma 7.5,  3 −  2 ∈ N Z 2 and hence, using also | | ≤ 4 √ N , Now it remains the case  2 =  3 . Such monomials cannot exist in H (1) in (3.16) since the monomials with two equal modes have been removed in (3.3) (it does not support degenerate rectangles). Naturally a degenerate rectangle may appear after we apply the change L (0) introduced in Theorem 4.4. Nevertheless, the map L (0) is identity plus smoothing (see statement 5 of that theorem), which leads to the needed N −1 factor. B1 σ 3 = σ 2 = +1, σ 1 = −1. Now the selection rule reads −  1 +  2 +  3 − v = 0, with again v = (− m · , 0). We rewrite (7.14) as By the first statement of Lemma 7.5, v −  2 , v −  3 = 0. By Property VIII Λ and the second statement of Lemma 7.5, one has |(  2 ,  3 )| ≥ N 2 and estimate (7.7) implies |  2 |, |  3 | ≤ N 3/2 .
and one concludes as in A1.
In conclusion we have proved that Item (i) of Lemma 5.3, jointly with estimate (7.16), implies that, for ρ ∈ (0, ρ/2] and r ∈ (0, r/2] This completes the proof of Lemma 7.6. The Hamiltonian G 0 in (7.11) is the Hamiltonian that the I-team derived to construct their toy model. A posteriori we will check that the remainder J 2 plays a small role in our analysis.
The properties of Λ imply that the equation associated to G 0 reads (7.17) iβ  = −β  |β  | 2 + 2β  child 1 β  child 2 β spouse + 2β parent 1 β parent 2 β  sibling for each  ∈ Λ. In the first and last generations, the parents and children are set to zero respectively. Moreover, the particular form of this equation implies the following corollary.
Corollary 7.7 ([CKS + 10]). Consider the subspace where all the members of a generation take the same value. Then, U Λ is invariant under the flow associated to the Hamiltonian G 0 . Therefore, equation (7.17) restricted to U Λ becomes The dimension of U Λ is 2g, where g is the number of generations. In the papers [CKS + 10] and [GK15], the authors construct certain orbits of the toy model (7.18) which shift its mass from being localized at b 3 to being localized at b g−1 . These orbits will lead to orbits of the original equation (2D-NLS) undergoing growth of Sobolev norms.
Theorem 7.8 ( [GK15]). Fix a large γ 1. Then for any large enough g and µ = e −γg , there exists an orbit of system (7.18), κ > 0 (independent of γ and g) and T 0 > 0 such that Moreover, there exists a constant K > 0 independent of g such that T 0 satisfies This theorem is proven in [CKS + 10] without time estimates. The time estimates were obtained in [GK15].

The approximation argument
In Sections 4, 5 and 6 we have applied several transformations and in Sections 6 and 7 we have removed certain small remainders. This has allowed us to derive a simple equation, called toy model in [CKS + 10]; then, in Section 7, we have analyzed some special orbits of this system. The last step of the proof of Theorem 1.1 is to show that when incorporating back the removed remainders (J 1 and R in (7.3) and J 2 in (7.10)) and undoing the changes of coordinates performed in Theorems 4.4 and 5.2, in Proposition 6.2 and in (7.1), the toy model orbit obtained in Theorem 7.8 leads to a solution of the original equation (2D-NLS) undergoing growth of Sobolev norms. Now we analyze each remainder and each change of coordinates. From the orbit obtained in Theorem 7.8 and using (7.19) one can obtain an orbit of Hamiltonian (7.11). Moreover, both the equation of Hamiltonian (7.11) and (7.18) are invariant under the scaling By Theorem 7.8, the time spent by the solution b ν (t) is where T 0 is the time obtained in Theorem 7.8. Now we prove that one can construct a solution of Hamiltonian (7.2) "close" to the orbit β ν of Hamiltonian (7.11) defined as where b(t) is the orbit given by Theorem 7.8. Note that this implies incorporating the remainders in (7.3) and (7.10). We take a large ν so that (8.3) is small. In the original coordinates this will correspond to solutions close to the finite gap solution. Taking J = J 1 + J 2 (see (7.3) and (7.10)), the equations for β and Y associated to Hamiltonian (7.2) can be written as Now we obtain estimates of the closeness of the orbit of the toy model obtained in Theorem 7.8 and orbits of Hamiltonian (7.2).
The proof of this theorem is deferred to Section 8.1. Note that the change to rotating coordinates in (7.1) does not alter the 1 norm and therefore a similar result as this theorem can be stated for orbits of Hamiltonian (6.3) (modulus adding the rotating phase).
Proof of Theorem 1.1. We use Theorem 8.1 to obtain a solution of Hamiltonian (3.13) undergoing growth of Sobolev norms. We consider the solution (Y * (t), θ * (t), a * (t)) of this Hamiltonian with initial condition Y * = 0 for an arbitrary choice of θ 0 ∈ T d . We need to prove that Theorem 8.1 applies to this solution. To this end, we perform the changes of coordinates given in Theorems 4.4, 5.2 and 6.2, keeping track of the 1 norm.
For L (j) , j = 1, 2, Theorems 5.2 and 6.2 imply the following. Consider (Y, θ, a) ∈ D(ρ, r) and define π a (Y, θ, a) := a. Then, we have This estimate is not true for the change of coordinates L (0) given in Theorem 4.4. Nevertheless, this change is smoothing (see Statement 5 of Theorem 4.4). This implies that if all  ∈ supp{a} satisfy | | ≥ J then Thanks to Theorem 7.3 (more precisely (7.7)), we can apply this estimate to (8.6) with J = Cf (g).
Using the fact that a * 1 ν −1 g2 g and the condition on ν in (8.5), one can check Therefore, we can conclude We define ( Y * , θ * , a * ) the image of the point (8.6) under the composition of these three changes. We apply Theorem 8.1 to the solution of (7.2) with this initial condition. Note that Theorem 8.1 is stated in rotating coordinates (see (7.1)). Nevertheless, since this change is the identity on the initial conditions, one does not need to make any further modification. Moreover, the change (7.1) leaves invariant both the 1 and Sobolev norms. We show that such solution ( Y * (t), θ * (t), a * (t)) expressed in the original coordinates satisfies the desired growth of Sobolev norms. Define To estimate the initial Sobolev norm of the solution (Y * (t), θ * (t), a * (t)), we first prove that The initial condition of the considered orbit given in (8.6) has support Λ (recall that Y = 0). Therefore, Then, taking into account Theorem 7.8, From Theorem 7.3 we know that for i = 3, Therefore, to bound these terms we use the definition of µ from Theorem 7.8. Taking γ > 1 2κ and taking g large enough, we have that a * (0) 2 h s ≤ 2ν −2 S 3 . To control the initial Sobolev norm, we need that 2ν −2 S 3 ≤ δ 2 . To this end, we need to use the estimates for ν given in Theorem 8.1, and the estimates for | | ∈ Λ and for f (g) given in Theorem 7.3. Then, if we choose ν = (f (g)) 1−σ , we have Note that Theorem 8.1 is valid for any fixed small σ > 0. Thus, provided s < 1, we can choose 0 < σ < 1 − s and take g large enough, so that we obtain an arbitrarily small initial Sobolev norm.
Remark 8.2. In case we ask only the 2 norm of a * (0) to be small we can drop the condition s < 1. Indeed a * (0) 2 ν −1 2 g g which can be made arbitrary small by simply taking g large enough (and ν as in (8.5)). Now we estimate the final Sobolev norm. First we bound a * (T ) h s in terms of S g−1 . Indeed, Thus, it is enough to obtain a lower bound for a *  (T ) for  ∈ Λ g−1 . To obtain this estimate we need to express a * in normal form coordinates and use Theorem 8.1. We split |a *  (T )| as follows. Define ( Y * (t), θ * (t), a * (t)) the image of the orbit with initial condition (8.6) under the changes of variables in Theorems 4.4 and 5.2, Proposition 6.2 and in (7.1). Then, The first term, by Theorem 7.8, satisfies |β ν  (T )| ≥ ν −1 /2. For the second one, using Theorem 8.1, we have a *  (T ) − β ν  (T )e iΩ  (λ,ε)T ≤ ν −1−σ . Finally, taking into account the estimates (8.7) and (8.8), the third one can be bounded as Now, by Theorem 8.1 and Theorem 7.3 (more precisely the fact that | | f (g) for  ∈ Λ), Thus, by (8.9), we can conclude that which, by Theorem 7.3, implies Thus, taking g large enough we obtain growth by a factor of K/δ. The time estimates can be easily deduced by (8.2), (8.5), (7.6) and Theorem 7.8, which concludes the proof of the first statement of Theorem 1.1. For the proof of the second statement of Theorem 1.1 it is enough to point out that the condition s < 1 has only been used in imposing that the initial Sobolev norm is small. The estimate for the 2 norm can be obtained as explained in Remark 8.2. 8.1. Proof of Theorem 8.1. To prove Theorem 8.1, we define We use the equations in (8.4) to deduce an equation for ξ. It can be written as (8.11) We analyze the equations for ξ in (8.10) and Y in (8.4).
Proof. Proceeding as forξ, we write the equation forẎ as We claim that X 1 (t) and X 1 (t) are identically zero. Then, proceeding as in the proof of Lemma 8.3, one can bound each term and complete the proof of Lemma 8.4. To explain the absence of linear terms, consider first ∂ βθ J (0, θ, β ν ). It contains two types of monomials: those coming from R 2 (see (4.5)) which however do not depend on θ, and those coming from R 4 (see (6.2)). But also these last monomials do not depend on θ once they are restricted on the set Λ (indeed the only monomials of R 4 which are θ dependent are those of the third line of (6.2), which are supported outside Λ). Therefore ∂ βθ J (0, θ, β ν ) ≡ 0 (and so ∂ βθ J (0, θ, β ν ) and ∂ Yθ J (0, θ, β ν )).
Since we are assuming (8.5) and we can take A large enough (see Theorem 7.3), we obtain that for t ∈ [0, T ], provided g is sufficiently large which implies that T ≤ T * . That is, the bootstrap assumption was valid. This completes the proof.
Appendix A. Proof of Proposition 6.1 We split the proof in several steps. We first perform an algebraic analysis of the nonresonant monomials.
A.1. Analysis of monomials of the form e iθ· a σ 1  1 a σ 2  2 a σ 3  3 a σ 4  4 . We analyze the small divisors (6.1) related to these monomials. Taking advantage of the asymptotics of the eigenvalues given in Theorem 4.6, we consider a "good" first order approximation of the small divisor given by Note that this is an affine function in ε and therefore it can be written as . We say that a monomial is Birkhoff non-resonant if, for any ε > 0, this expression is not 0 as a function of λ.
Lemma A.1. Assume that the m k 's do not solve any of the linear equations defined in (A.5) (this determines L 2 in the statement of Theorem 6.1). Consider a monomial of the form e iθ· a σ 1 j 1 a σ 2 j 2 a σ 3 j 3 a σ 4 j 4 with (j, , σ) ∈ A 4 . If (j, , σ) ∈ R 4 , then it is Birkhoff non resonant.
If = 0, we have r σ r n r = r σ r n 2 r = 0. Then {n 1 , n 3 } = {n 2 , n 4 }. One verifies easily that in such case the sites  r 's form a horizontal trapezoid (that could be even degenerate).
A.2. Analysis of monomials of the form e iθ· Y l a σ 1 j 1 a σ 2 j 2 . In this case, since the factor Y l does not affect the Poisson brackets, admissible monomials (in the sense of Definition 4.2) are non-resonant provided they do not belong to the set R 2 introduced in Definition 4.3.
Lemma A.2. Any monomial of the form e iθ· a σ 1  1 a σ 2  2 Y i with (j, , σ) / ∈ R 2 admissible in the sense of Definition 4.2 is Birkhoff non-resonant.
Proof. We skip the proof since it is analogous to Lemma 6.1 of [MP18].
We can now prove the following result.
for some constant γ depending on ε .
Proof of Proposition A.5. If the integer K is sufficiently large, namely |K| ≥ 4 | | max So from now on we restrict ourselves to the case |K| ≤ 4 | | max 1≤i≤d (m 2 i ). We will repeatedly use the following result, which is an easy variant of Lemma 5 of [Pös96].
The proof relies on the fact that all the functions appearing in (A.10) are Lipschitz in λ, for full details see e.g. Lemma C.2 of [MP18]. Now, let us fix (A.11) γ = ε M 0 100 .
Estimate (A.15) gives immediately which is what we claimed.
We can finally prove Proposition 6.1.