Normal forms \`a la Moser for aperiodically time-dependent Hamiltonians in the vicinity of a hyperbolic equilibrium

The classical theorem of Moser, on the existence of a normal form in the neighbourhood of a hyperbolic equilibrium, is extended to a class of real-analytic Hamiltonians with aperiodically time-dependent perturbations. A stronger result is obtained in the case in which the perturbing function exhibits a time decay.


INTRODUCTION
The classical theorem of Moser, proven in [Mos56], establishes the existence of a (convergent) normal form in a neighbourhood of a hyperbolic equilibrium of an area preserving map, either autonomous or periodically dependent on time. A result contained in [CG94], extends this result to the the flow of a priori unstable system in a neighbourhood of a partially hyperbolic torus, including in this way the quasiperiodic case. A concise description of the latter case can be found in [Gal97]. The aim of this paper is to show the existence of a normal form for Hamiltonians in the form (1), i.e. real-analytic and non-autonomous perturbations of a hyperbolic equilibrium, for which the time dependence is not required to be periodic or quasiperiodic i.e. aperiodic. In the same spirit of the aperiodic version of the Kolmogorov theorem of [FW14a], which we use as a guideline (see also [Giob]), the proof consists on the extension of the KAM approach of [CG94] and [Gal97]. Even in the original problem of Moser, despite the absence of "genuine" small divisors 1 , the well known property of superconvergence of the KAM schemes, turns out to be of crucial importance in order to compensate the accumulation of "artificial" divisors generated by the Cauchy estimates. This feature is profitably used also in our case. The treatment of the class of time-dependent homological equations, naturally arising in the normalization algorithm, has been improved with respect to [FW14a]. Basically, the canonical transformation on which the single step of the mentioned algorithm is based, has the property to leave the time unchanged 2 . Hence, this can be interpreted as a family of canonical maps for which the time plays the role of "parameter". This allows to weaken the analyticity hypothesis for the time dependence leading to a remarkable simplification of the quantitative estimates. The proof is carried out by using the formalism of the Lie series method developed by Giorgilli et al. (see e.g. [Gio03] and references therein).
The function F will be supposed to be real-analytic in p and q and such that, denoted as f α (t) its Taylor coefficients, one has f α (t) = 0 for all 3 |α| ≤ 2, and all t ∈ R + . Namely, the Taylor expansion of F starts from the terms of degree 3. The standard framework for the analysis, features the complexification of the domain D as follows. Let R ∈ (0, 1/2] and define The perturbation F will be supposed continuous on Q R and holomorphic in the interior for all t ∈ R + (then H is on D R ) for some R. It will be sufficient to suppose that the real and imaginary parts of the complex valued functions f α (t) belong to C 1 (R + ) for all α. Given a function G : Q R × R + → C, we consider the Taylor norm where | · | + := sup t∈R + | · |. Clearly |G| R := sup Q R |G| + ≤ G R . We briefly recall the following standard result (which motivates the above described assumptions on F ): if a function G is continuous on Q R and holomorphic in the interior, for all t ∈ R + , one has |g α (t)| + ≤ |G| R R −|α| . In particular, G R ′ < +∞ for all R ′ < R. In the described setting the main result can be stated as follows Exactly as in the classical Moser theorem, the quantity x (∞) is a first integral, hence the flow associated to Hamiltonian (3) can be reduced to quadratures. In particular, one has The use of an additional ingredient leads to an even stronger result. Given G : Q R × R + → C we define as the "time-dependent" Taylor norm of G, the quantity G R;R + := α |g α (t)|R |α| , i.e. (2) in which | · | + is replaced with | · |. Now we introduce the next Hypothesis 2.2. (Slow decay) Suppose that there exist M F ∈ [1, +∞) and a > 0 such that for all (p, q, t) ∈ Q R × R + .
In this way we are able to prove the following The Hypothesis 2.2, already used in [FW14a], turns out to be necessary in order to ensure the existence of certain improper integrals, which appear when dealing with time-dependent homological equations. As in the latter paper, this particular rate of decay is assumed only for simplicity of discussion. Similarly, we stress that no lower bounds are imposed on a (except zero), in this way the time decay can be arbitrarily slow. The natural side-effect is that the estimates on the convergence radius of the normal form worsen as a is smaller and smaller. It should be stressed that, in both cases, the choice of ω in the interval (0, 1] is discussed as the "interesting" case. On the other hand, it is clear that the contribution of the time perturbation is smaller as ω increases 4 . That is why, the case ω ≥ 1 can be treated with the same tools leading, in general, to easier estimates. The proof of Theorem 2.1 is (traditionally) achieved in two steps. In the first one (Sec. 3), a suitable normalization algorithm is constructed and discussed at a formal level. In the second part (Sec. 5) the problem of its convergence is addressed, after having stated some tools of a technical nature (Sec. 4). Proof of Theorem 2.3 is just a variazione sul tema. The necessary modifications are outlined in Sec. 6.

THE FORMAL PERTURBATIVE SETTING
The formal perturbative algorithm has the typical inductive structure. To start, we shall suppose that Hamiltonian (1) can be written at the j−th stage of the normalization process as withF (j) at least of degree 3 in p, q. It is immediate to realize that (1) is in the form (6) so that we can set H (0) := H. Our aim is to construct a class of canonical transformations M j , parametrised by t, such that H (j+1) := H (j) • M j is still of the form (6). Roughly, the transformations M j will be determined in such a way the "mixed" terms, i.e. of the form p α 1 q α 2 with α 1 = α 2 contained in the perturbation, are "gradually" removed as j increases, while the terms of the form (pq) n are progressively stored inJ (j) . This effect will be quantified in the next section, showing that the size of the "residual" perturbation is asymptotic to zero, as j → ∞. Hence one sets so that, at least formally, First of all we writẽ where F (j) contains only "mixed" terms. Now we consider the action on H (j) of the transformation M j , which is defined by the the Lie series is the (unknown) generating function. Supposing that it is possible to determine it in such a way one has that, by settingJ (j+1) := J (j) , and the transformed Hamiltonian H (j+1) := exp(L χ (j) )H (j) has exactly the form (6). Note that by (10) and (11) Defining g (j) (x, t) := ∂ x J (j) (x, t) one has that equation (10) reads as having denoted ð := q∂ q − p∂ p . Taking into account of the expansion F (j) =: the solution of equation (13) reads as α,0 (x) are functions to be determined. Clearly, we shall set F (j) α,0 (x) ≡ 0 for all α such that α 1 = α 2 and such that f α (x, t) are identically zero for those values. It is evident that as |α| ≥ 3 for by hypothesis on F (j) , the generating function χ (j) will be at least of degree 3. This implies that, by (12),F (j+1) will be at least of degree 4, in particular it will not contain terms of degree 2. By hypothesis on F ≡ F (0) and by induction, this is true for all j, implying that g (j) (0, t) = ω for all t ≥ 0, i.e. g (j) has a strictly positive real part (by hypothesis on ω), in a suitable neighbourhood of the origin and more precisely via a suitable choice of R 0 . This will play a crucial role in our later arguments. The formal part is complete.
Remark 3.1. It is immediate to recognize the similarity between equation (13) and those found in [FW14a] and [FW14b]. The main difference is the presence of the function g (j) (x, t) which requires a careful analysis about its variation on time, as anticipated above.

Bounds on the solutions of the homological equation.
First of all let us recall the following elementary equality, valid for all λ ∈ [0, 1), which will be repeatedly used in the follow Then we state the next and that, for all (x, t) ∈ Q R j × R + one has Then for all δ ∈ (0, 1) the solution of (13) satisfies Remark 4.2. Note that hypothesis (17a) is essential as it is easy to find g (j) (x, t) satisfying (17b) for which the solution of (14) is unbounded on R + .
The proof goes along the lines of a similar result contained in [FW14a], with the remarkable simplification due to the fact that now t is purely real. The very minor drawback with respect to the "analytic" case treated in [FW14a], is that, in this case, the estimate of the time derivative does not follow directly from a Cauchy estimate.

An estimate on the Lie operator.
This is a standard result in the works of A. Giorgilli et al., see e.g. [Gioa]. The statement recalled below, is adapted to the notational setting at hand Lemma 4.3. Suppose that χ (1−δ)R and G (1−δ)R are bounded for some δ ∈ (0, 1/2). Then We shall also consider the case of bounded G R , for which (22) clearly holds with the obvious replacement. It is evident that a sufficient condition for the convergence of the Lie operator exp(L χ ) is that e 2 δ −2 χ (1−δ)R ≤ 1/2.

QUANTITATIVE ESTIMATES
5.1. The iterative lemma. Let us consider a sequence {u (j) } j∈N ∈ [0, 1] 5 with u (0) to be determined, where u (j) := (d j , ε j , R j ,m j ,M j ). Let u * := (0, 0, R * ,m * ,M * ) with ω/2 ≤m * <M * ≤ (3/2)ω and R * > 0 to be determined as well. Our aim is now to prove the next Lemma 5.1. Suppose that for some j ∈ N, there exists u (j) with u (j) l > (u * ) l for l = 1, . . . , 4 and M j <M * , satisfying for all (x, t) ∈ Q R j × R + . Then, under the condition it is possible to determine u The validity of (24) and (25) (compare with (17a) and (17b)) with the above mentioned bounds onm * and onM * , is clearly related to the possibility of using Prop 4.1 for all j .
Proof. First of all, immediately from (8) and (23), it follows F (j) R j ≤ ε j . On the other hand, recall The last quantity is well defined as a consequence of the (stronger) condition (26), beingm j >m * ≥ ω/2. Similarly, |g (j) (x, t)| ≤M j + ε j (R * d j ) −2 =: M j . From Lemma 4.3 with δ = d j , (12), (18) and (23), under the convergence condition guaranteed by (26) we get Hence we shall set in order to obtain the validity of (23), (24) and (25) at the j + 1-th step. The first of (29) is the well known "heart" of the quadratic method.

Determination of the bounding sequences.
Our aim is now to construct the sequence u (j) for all j under the constraints (29) and show that lim j→∞ u (j) = u * . The last step will be the determination of u 0 , completed in the next section. The procedure is analogous to [FW14a]. We start by choosing, for all j ≥ 1 ε j := ε 0 (j + 1) −12 .

AN OUTLINE OF THE PROOF OF THEOREM 2.3
In this section we describe the necessary modifications in the proof of Thm. 2.1 in order to get its "strong" version. However, we stress that the crucial point is the following: if we suppose the existence of the integral R + f (j) α (t)dt (guaranteed by the exponential decay of F (j) ), then (14) exists on R + also for λ = 0 i.e. the r.h.s. of the homological equation can contain also terms with α 1 = α 2 .
Formal scheme. The definition ofJ (j) and ofF (j) is not necessary, we suppose that H (j) is directly of the form H (j) = ωpq + η + F (j) (p, q, t). (38) The initial Hamiltonian is exactly of the form above, so we can set H (0) := H. Suppose that χ (j) is chosen in a way to satisfy the homological equation it is sufficient to define in order to have H (j+1) of the form (38). By expanding χ (j) = α c α (t)p α 1 q α 2 and F (j) as well 6 , we get this time, for all αċ withλ := ω(α 1 − α 2 ) purely real. 6 Note that in this case the Taylor expansion of F (j) will contain also terms with α1 = α2.