On symplectomorphisms and Hamiltonian flows

We propose the construction of a sequence of time one flows of autonomous Hamiltonian vector fields, converging to a fixed near the identity C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^1$$\end{document} symplectic diffeomorphism ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document}. This convergence is proved to be uniformly exponentially fast, in a non analytic symplectic topology framework.


Introduction
The aim of this note is to revisit the classical issue to find in correspondence of any fixed small (i.e. close to identity) symplectic diffeomorphism ψ a sequence of compositions of time one flows Φ 1 XF j of autonomous Hamiltonian vector fields X Fj , and to discuss its possible convergence to ψ. This subject is crucial in the C ω Hamiltonian perturbation theory. There is a long history around this matter. After early pioneering papers [5,6], a rigorous setting has been provided mainly in [8,9], and other interesting references therein quoted. More recently, Giorgilli [10] introduces in a genuinely innovative way a 'Lie transform', generalizing the Lie series and having a number of nice algebraic properties; this object appears as a useful tool which could give new help to a greater comprehension of the Baker-Campbell-Hausdorff (BCH) matter. Unlike the analytic framework of the above quoted papers, the novelty in the present work is to act in a C 1 environment for diffeomorphisms ψ, possibly for small compact perturbations of the identity, which are typical in many questions of symplectic topology. Our construction will lead to an infinite product converging uniformly exponentially fast to the fixed ψ: It will be provided by a global realization of Weinstein's neighbourhood theorem in T * R n proposed by Viterbo (see [4,20]), and the exponential estimate (1) will be produced by means of standard facts from mid-point approximation theory. This results suggests that, by means of an adequate version of the BCH theorem, one could solve the problem of the construction of an autonomous Hamiltonian vector field X F whose time-one flow is a very fine approximation of an assigned symplectic diffeomorphism ψ, all this in a non analytic category. The effort to obtain anyway some good approximations in such a direction has been made inside the community of the Hamiltonian perturbation theory, see the discussion in [2] and also [13] for allied topics in the analytic category. As a matter of fact, the paper [2] by Benettin and Giorgilli is a point of arrival in the analytic realm: the authors show that ε-small symplectic diffeomorphisms can be asymptotically, up to terms of order O(exp(−ε * /ε)), approximated by autonomous Hamiltonian time one flows.
is a subgroup of Diff ω,0 (M ) called Ham(M, ω). As Polterovich and Rosen point out [17], although in general the inclusion Ham(M, ω) ⊂ Diff ω,0 (M ) is strict, the difference between the two groups is 'not too big'. Actually, when ω = dϑ, the gap between Ham(M, ω) and Dif f ω,0 (M ), is essentially cohomological one, and, e.g. as in [21], this fact can be seen directly in a clear way: on one hand, given ψ ∈ Diff ω,0 (M ), we have that ψ * ϑ − ϑ is closed, Vol. 24 (2022) On symplectomorphisms and Hamiltonian flows Page 3 of 12 33 so that is exact. In the cotangent bundle case, M = T * Q with ϑ = pdq, the generating function in the r.h.s. of (5) becomes the well known and we have seen right now that Banyaga [1] showed that actually (see also [16] Sec. 14.1, and [21] Ex. 6.3 p. 23), since ω is exact, In a topologically trivial environment-like T * R n , where H 1 = {0} -Banyaga's result tells us that Diff ω,0 (M ) = Ham(M, ω), in other words, any symplectomorphism is the time one flow for some possibly time dependent Hamiltonian; this fact has been known already for a long time inside the specific world of the Hamiltonian perturbations theory- [5], [15]-, as we can read e.g. in [2]. The time dependence is crucial, because in such a case Ham(M, ω) turns out to be even a subgroup of Dif f ω,0 (M ): the composition of several time one flows related to (even time independent) Hamiltonian vector fields is a time one flow, for some, possibly time dependent, Hamiltonian vector field.
In the next Sections we revisit the problem of the approximation of the symplectomorphisms by time one flows related to time independent Hamiltonian systems. As already said, this subject has long been studied in the context of the theory of perturbation of Hamiltonian systems. We tackle this matter by investigation of simple techniques borrowed from symplectic topology. More precisely, this note would represent an attempt to realise some first steps towards a perturbative symplectic topology scheme; in other words, we direct our attention towards reaching perturbation results starting from a C 1 context, even though iterated Lie brackets or vector fields do force us immediately in a C ∞ and then Gevrey environment, well adequate to consider compactly supported objects, even though the first result, see (43) and (45), is a purely topological one. By the way, we find along our road map a quadratic estimate-see (35)-evoking an analogue one arising in Hamiltonian perturbation theory, producing in that context well known Newton-like efficient approximation algorithms.

Symplectomorphisms as Lagrangian submanifolds
The following Lemma is a summary of Weinstein's Lagrangian neighborhood theorem (see [22], and Prop. 3.4.13 -14 of [14]) in a form that fits our needs. and Here and inafter, by ω M we mean the standard symplectic 2-form on T * M . As a consequence of this Lemma, we have the next fact. Let ψ ∈ Dif f ω,0 (M ) be a symplectomorphism sufficiently C 1 -close to the identity, then the image by Ψ of the a Lagrangian submanifold graph(ψ) is Lagrangian in T * M which is candidate to be transverse to the fibers of π M . When M is topologically trivial, like M = T * R n = R 2n , versions of this Lemma have been utilised in some directions. The following ingenious linear symplectomorphism by Viterbo [20] realises explicitly the above task: We have denoted the standard projections by T * R n pr1 ←− T * R n × T * R n pr2 −→ T * R n and we may easily verify the main property f * (ω T * R n ) = Ω. This transformation has been e.g. utilized in [3], other authors, as Traynor [19] and Sandon [18], introduced the alternative symplectic map at first glance simpler, but not possessing the precious property leading us to the midpoint finite reduction of Hamiltonian systems, see Theorem 3.1 right below.
Coming back to our canonical transformation or symplectomorphism ψ ∈ C 1 (T * R n ; T * R n ), let us denote by its deviation from the identity, which we suppose compactly supported in T * R n , and we ask that Vol. 24 (2022) On symplectomorphisms and Hamiltonian flows Page 5 of 12 33 Set we notice that f (graph(ψ)), is a deformation of the null section, and also a Lagrangian submanifold Λ of T * (T * R n ). Indeed graph(ψ) is a Lagrangian submanifold (Ω| graph(ψ) = 0) and f is a symplectomorphism, hence ω T * R n f (graph(ψ)) = 0.
By denoting as usual J = O I −I O , we write the inclusion of this Lagrangian submanifold into T * (T * R n ), is a diffeomorphism of T * R n into itself, which is a small deformation of the identity. Indeed, it is a perturbation of the identity by the map 2 which is contractive by (16) : Lip( 2 ) < 1. This fact is telling us that f (graph(ψ)) is globally transverse to the fibers of π T * R n . Hence, there exists a generating function F ∈ C 2 (T * R n ; R) for it (without Maslov-Hörmander auxiliary parameters, see e.g. [4,12,23]), such that linking (18) with (23) by the diffeomorphism Actually, from (26) we see that the generating function F can be interpreted as a genuine time-independent Hamiltonian function: more precisely, the above discussion can be summarized in the following Theorem 3.1. Let ψ be a C 1 (T * R n ; T * R n ) symplectic diffeomorphism satisfying Lip(ψ − id) < 2 with ψ − id compactly supported-see (16). Then there exists a time independent Hamiltonian function F : R 2n → R such that ψ is exactly the solution ϕ 1 XF of the step one Euler midpoint 1 , related to the Hamiltonian vector field X F = JdF : that is, see (26): ψ = ϕ 1 XF . Note that so far the compact support hypothesis has not yet been used: this will instead be essential in the next Section to guarantee the completeness of the flows.

Setting and definitions
We define In view of the iteration implemented in the next Sect. 4.2, we will insert the 'zero' subscript into the mathematical objects introduced above. From (25) we see that Moreover we define F 0 , The hypothesis made above that ψ − id be compactly supported in T * R n assures us that the C 1 Hamiltonian vector field X F0 admits a complete flow Φ t XF 0 . To look for time one flow of X F0 is equivalent to look for time h 0 flow of X F0 : and analogously-by (31)-for the Euler midpoint: Vol. 24 (2022) On symplectomorphisms and Hamiltonian flows Page 7 of 12 33 Note that if d := diam supp(ψ − id), by asking F 0 | R 2n \supp (ψ−id) ≡ 0 (normalization), then

Main estimate and its iterations
Given ψ, by (25) we get a function F 0 such that ψ is exactly the step one Euler midpoint representation of the flow for X F0 , ψ = ϕ 1 . By using the Euler midpoint approximation settled in Section A, the deviation (35) We will iterate this procedure: instead of ψ, in the next step we consider the symplectomorphism ψ • Φ −1 XF 0 , uniformly closer to the identity than ψ. Preliminarily, recall that, for any function g : R m → R m with |g − id| C 0 < +∞ and any diffeomorphism f : R m → R m , we have that (36) Coming back to our task: In the Sect. 3 above we asked on ψ, a given 'first order' diffeomorphism near the identity, the requirement (17) precisely to guarantee that 1 2 (id + ψ) is a diffeomorphism too. Differently, now we observe that the composition ψ • Φ −1 XF 0 is 'second order' diffeomorphism near to the identity, so we claim that the analogous 1 2 (38) The global transversality with respect to the fibers of π T * R n is offering the existence of a new generating function F 1 : R 2n → R, such that We rewrite the relations (35) for this second step (note that (42) By iterating k-times this procedure, we get Furthermore, recalling (25), In other words, we achieve a sequence of time one Hamiltonian flows {Φ 1 XF k } k∈N uniformly convergent (and highly fast) to the assigned ψ: Here below, we list the hierarchy of the steps giving the sequence, where we denote by ϕ 1 XF k the solution of the step one Euler midpoint difference reduction related to X F k : Obviously, (45) does work if |ψ − id| C 0 = h 0 < 1, so that, together with condition (16), we could summarize our requirements by The above discussion can be summarized in the following Theorem 4.1. Let ψ be a C 1 (T * R n ; T * R n ) symplectic diffeomorphism satisfying |ψ − id| C 1 < 1 with ψ − id compactly supported. Then there exists a sequence of time independent Hamiltonian function F k : R 2n → R, k ∈ N, such that ψ is represented by the uniform composition limit (45).