An alternative proof of infinite dimensional Gromov's non-squeezing for compact perturbations of linear maps
aa r X i v : . [ m a t h . S G ] O c t An alternative proof of infinite dimensionalGromov’s non-squeezing for compactperturbations of linear maps
Lorenzo Rigolli ∗ October 30, 2019
Abstract
This paper deals with the problem of generalising Gromov’s nonsqueezing theorem to an infinite dimensional Hilbert phase space set-ting. By following the lines of the proof by Hofer and Zehnder offinite dimensional non-squeezing, we recover an infinite dimensionalnon-squeezing result by Kuksin [Kuk95a] for symplectic diffeomor-phisms which are non-linear compact perturbations of a symplecticlinear map. We also show that the infinite dimensional non-squeezingproblem, in full generality, can be reformulated as the problem of find-ing a suitable Palais-Smale sequence for a distinguished Hamiltonianaction functional. ∗ This work was partially supported by the DFG grant AB 360/1-1.E-mail: [email protected] . ntroduction A strong symplectic form on a real Hilbert space H is a skew-symmetriccontinuous bilinear form ω : H × H → R , which is strongly non-degenerate, namely the associated bounded linear op-erator Ω : H → H ∗ defined by < Ω x, y > = ω ( x, y ) ∀ x, y ∈ H , where < · , · > denotes the duality pairing, is an isomorphism. A Hilbert space ( H , ω ) endowed with a strong symplectic form ω is said symplectic Hilbertspace . In this paper we study the global 2-dimensional rigidity phenomenongiven by Gromov’s non-squeezing, in the setting of infinite dimensional sym-plectic Hilbert spaces.As first noticed by Zakharov [Zak74], the Hamiltonian formalism is useful notonly to study classical mechanics but also some evolutionary PDEs that sat-isfy the conservation of energy. The solutions of these PDEs are given by theHamiltonian flow associated to a scalar function defined on a distinguishedinfinite dimensional symplectic space, which is usually uniquely determinedby the PDE under consideration.Just to make an example, under suitable assumptions on the nonlinearity f ,the nonlinear wave equation: u tt − ∆ u + f ( u ) = 0 , for u : R × T n → R , where T n is the n -torus, defines a Hamiltonian flowon the Sobolev space H ( T n ) × H ( T n ) which carries a strong symplecticform. For infinite dimensional Hamiltonian systems like this it makes senseto speculate about the validity of non-squeezing. (Infinite dimensional squeezing question) Let ( H , ω ) be a symplecticHilbert space with a compatible inner product. For < s < r , is it possible2or a Hamiltonian flow to map a ball of radius r in H into a cylinder givenby all vectors of H whose symplectic projection onto a plane V has distanceless than s from a given vector in V ?As first observed by Kuksin [Kuk95a], the non-squeezing property for non-linear Hamiltonian PDEs would have relevant physical implications; for in-stance in the transfer of energy problem of understanding whether the energyof nonlinear conservative oscillations spreads to higher frequencies. Moreovernon-squeezing would also imply uniformly asymptotically stability for solu-tions of Hamiltonian PDEs.Prompted by these motivations Kuksin showed that the non-squeezing holdsif the flow of the PDE is a compact perturbation of a linear one and sometechnical assumptions are fulfilled [Kuk95a]. Not all Hamiltonian PDEs havea flow of this form, however under certain conditions this is the case for non-linear hyperbolic equations like the nonlinear versions of the wave equation,the Schrödinger equation, the membrane equation and the string equation.The non-squeezing property was later proved also for some equations whoseflow is not a compact perturbation of a linear one, like the cubic Schrödingerequation considered by Bourgain [Bou94b], the KdV equation considered byColliander, Keel, Staffilani, Takaoka and Tao [CKS+05], and the BBM equa-tion considered by Roumégoux [Rou10]. This was done using quite delicateapproximations of the flow with a finite dimensional one and then applyingstandard Gromov’s non-squeezing.At this point we remark that two different notions of symplectic form arecommonly used in the Hilbert space setting, namely strong symplectic formsand weak symplectic forms. Strong symplectic forms are usually defined onSobolev spaces of functions with low regularity and in this case it makes senseto investigate the validity of non-squeezing; anyway it is important to noticethat if the phase space associated to the PDE is a Hilbert space of smootherfunctions which carry only a so-called weak symplectic form (namely a closednon-degenerate 2-form), then the squeezing is possible and energy transfer is3ypical, see for example [Kuk95b] and [CKS+10].A natural approach to deal with the infinite dimensional squeezing ques-tion is extending well known symplectic capacities to infinite dimensionalsubsets, since the existence of a normalized symplectic capacity implies thenon-squeezing. In this direction, Abbondandolo and Majer [AM15] used theconvex geometry tool given by Clarke’s duality in order to construct an in-finite dimensional symplectic capacity for convex sets and thus to prove anon-squeezing theorem when the image of the unit ball under a symplecto-morphism is a convex set.The main goal of this paper is to prove an infinite dimensional non-squeezing result which slightly generalizes the result by Kuksin in [Kuk95a].In order to give a precise statement we briefly recall some basic conceptsabout symplectic structures on infinite dimensional separable Hilbert spaces(for more details see Chapter 2 of [AM15] or [CM74], in which a more generalBanach setting is considered).Let ( H , ω ) be a symplectic Hilbert space; the choice of a Hilbert inner product h· , ·i on H determines a bounded linear operator J : H → H such that h J x, y i = ω ( x, y ) ∀ x, y ∈ H . Being the composition of Ω by the isomorphism H ∗ ∼ = H induced by the innerproduct, J is also an isomorphism. The skew symmetry of ω now reads as J T = − J , where J T : H → H is the adjoint operator with respect to theinner product.We say that an inner product h· , ·i is compatible with ω if one of the followingconditions (which are actually equivalent) holds1) Ω is an isometry (where H ∗ is endowed with the dual norm),2) J is an isometry,3) J is a complex structure (i.e. J = − I ).4ne can show that every symplectic Hilbert space ( H , ω ) admits a compatibleinner product h· , ·i . In general there is not a unique inner product compatiblewith a fixed ω , nevertheless the unit balls corresponding to different compat-ible inner products are all linearly symplectomorphic.Let ( H , ω ) be a symplectic Hilbert space endowed with a compatible in-ner product h· , ·i and let { e i , f i } i ∈ N be a countable orthonormal basis suchthat any { e i , f i } spans a symplectic plane. For any n ∈ N we consider theorthogonal projections P n : H → H n x n X i =1 h x, e i i e i + h x, f i i f i , (1)onto the n -dimensional symplectic Hilbert subspace H n .Let us consider the group of admissible symplectomorphisms Symp a ( H , ω, h· , ·i ) := ¶ ϕ ∈ Symp ( H , ω ) (cid:12)(cid:12)(cid:12) for k = ± , Dϕ k and D ϕ k are bounded ;( I − P n ) ϕ k | P n H −→ n → + ∞ uniformly on bounded sets ;[ P n , Dϕ k ( x ) ∗ ] −→ n → + ∞ in operators’ norm, uniformly in x ∈ H on bounded sets © . We will prove the following infinite dimensional non-squeezing result.
Theorem (Infinite dimensional non-squeezing) . Let ( H , ω ) be a Hilbert sym-plectic space endowed with a compatible inner product h· , ·i , B r the ball centredin with radius r and Z R a cylinder whose basis lays on a symplectic planeand has symplectic area πR . Let ϕ ∈ Symp a ( H , ω, h· , ·i ) , if ϕ ( B r ) ⊂ Z R then r ≤ R . We will see that the theorem above applies to symplectomorphisms whichare compact perturbations of linear maps, under slightly less restrictive as-sumptions than the ones considered in [Kuk95a].In view of applications we remark that the condition we require on the bound-edness of the differentials of ϕ ∈ Symp a ( H , ω, h· , ·i ) is not very restrictive, in5act the differentials of any order of the flow of a typical Hamiltonian PDEwhich is well-posed on its phase space are bounded on bounded sets.To prove the theorem we follow the approach adopted by Hofer and Zehnderin order to deduce the non-squeezing theorem for symplectomorphisms of ( R n , ω ) , see [HZ94]. What they did is observing that the non-squeezing isimplied by the existence of an appropriate critical point for a distinct Hamil-tonian action functional and then they proved the existence of such a criticalpoint by applying a minimax argument to the action functional.The main difference in dealing with infinite dimensional spaces instead ofwith Euclidean spaces is that we face a loss of local compactness: this hastwo major manifestations.First, unlike in the finite dimensional case, the Hamiltonian action functionalwe are interested in does not necessarily satisfy the Palais-Smale conditionand so we may find (PS) sequences which do not converge to a critical point.Therefore, in the infinite dimensional setting, (PS) sequences of the actionfunctional play the role which in the Hofer-Zehnder setting is played by crit-ical points. Secondly, in the infinite dimensional setting, in order to be ableto determine a (PS) sequence for the action functional we may have to askfor an additional compactness property. This compactness can be recoveredby restricting the class of symplectomorphisms under consideration to theone of admissible symplectomorphisms.We work with admissible symplectomorphisms only in order to find suitablePalais-Smale sequences, but as we will show, the fact that the infinite di-mensional non-squeezing question can be reduced to a purely critical pointtheory problem holds without imposing any restriction to the set of symplec-tomorphisms under consideration.The paper is organized as follows.In Section 1 we introduce the analytical setting in which we will work.6n Section 2 we characterize (PS) sequences of the Hamiltonian action func-tional and we show that they are preserved by symplectomorphisms as wellas their corresponding almost critical levels.In Section 3 we prove the infinite dimensional non-squeezing theorem via aminimax argument in combination with a finite dimensional approximationtechnique and we notice that this generalizes the non-squeezing theorem byKuksin.In Section 4 we show how, in bigger generality, the infinite dimensional non-squeezing problem can be formulated as a critical point theory problem. Acknowledgments.
I would like to warmly thank Alberto Abbondan-dolo for all the precious help and advice he gave me concerning this paper.
Let ( H , ω ) be a separable symplectic Hilbert space endowed with a com-patible inner product. On a countable orthonormal basis { e i , f i } i ∈ N the com-plex structure is defined by J ( e i ) = f i ,J ( f i ) = − e i , for any i . Given any smooth time-independent Hamiltonian function H ∈ C ∞ ( H , R ) , the associated Hamiltonian action functional A H : C ∞ ( S , H ) → R can be written as A H ( x ) = − Z h J ˙ x ( t ) , x ( t ) i dt − Z H ( x ( t )) dt, where C ∞ ( S , H ) is the space of smooth loops with values in H .For our purposes it is more convenient to deal with a Hilbert space than withthe space of smooth loops, thus we observe that any loop x ∈ C ∞ ( S , H ) isan element of L ( S , H ) which can be represented by its Fourier series as x ( t ) = X k ∈ Z e πkJt x k , x k ∈ H which are defined by the formula x k := Z e − πktJ x ( t ) dt, k ∈ Z . The first term of the action functional describes its symplectic part a ( x, y ) := − Z h J ˙ x ( t ) , y ( t ) i dt, while the second one describes the part depending on the Hamiltonian b H ( x ) = Z H ( x ( t )) dt. Inserting the Fourier expansion of x, y ∈ C ∞ ( S , H ) into a ( x, y ) and noticingthat Z h e πjJt x j , e πkJt y k i dt = δ jk h x j , y k i , we obtain that a ( x, y ) = 2 π X j ∈ Z j h x j , y j i = 2 π X j> | j |h x j , y j i − π X j< | j |h x j , y j i . Let us recall that for s non-negative, the fractional Sobolev spaces are definedas H s ( S , H ) := { x ∈ L ( S , H ) (cid:12)(cid:12)(cid:12) ∞ X k ∈ Z | k | s k x k k < ∞} . The standard inner product on the Hilbert space H s ( S , H ) is given by h x, y i s := h x , y i + 2 π X k =0 | k | s h x k , y k i , and the induced norm is | x | s := » h x, x i s . Therefore, by the previous remark the functional a can be extended to thespace H ( S , H ) := { x ∈ L ( S , H ) (cid:12)(cid:12)(cid:12) ∞ X k ∈ Z | k |k x k k < ∞} , H .On H ( S , H ) it is possible to define another equivalent norm. The Slo-bodeckij semi-norm of an element x ∈ H s ( S , H ) with < s < is thequantity | x | sem,s := Ä Z S Z S | x ( t ) − x ( r ) | | t − r | s dtdr ä . This semi-norm induces a norm on H s ( S , H ) | x | ′ s := | x | L + | x | sem,s , such that there exist constants C , C ≥ for which C | x | ′ s ≤ | x | s ≤ C | x | ′ s , for any x ∈ H s ( S , H ) .To keep the notation shorter we denote E := H ( S , H ) . Let us consider the orthogonal spitting E = E − ⊕ E ⊕ E + , into the spaces having non vanishing Fourier coefficients only for k < , k = 0 and k > respectively.If we denote by P − , P and P + the orthogonal projections onto the linearsubspaces E − , E , E + , we get that any element of E can be written as x = x − + x + x + , where P − ( x ) = x − , P ( x ) = x and P + ( x ) = x + .Summarizing the previous discussion, we saw that the map a ( x, y ) = 12 h ( P + − P − ) x, y i = 12 h x + , y + i − h x − , y − i
9s a continuous bilinear form on E , whose associated quadratic form a ( x ) = a ( x, x ) = 12 | x + | − | x − | is negative definite on E − , positive definite on E + and has E as kernel.Its differential is given by da ( x )[ y ] = h ( P + − P − ) x, y i , therefore its gradient with respect to the inner product on E is ∇ a ( x ) = ( P + − P − ) x. Before proceeding with the study of the action functional A H it is useful tostate a couple of general results regarding fractional Sobolev spaces. Proposition 1.1.
The space H ( S , H ) is continuously embedded into L p ( S , H ) for any ≤ p < ∞ . More precisely there exist a constant C p such that | x | L p ≤ C p | x | , for any H ( S , H ) .Proof. Using the trace theorem the result can be reduced to an application ofthe Sobolev’s embedding theorem for integer Sobolev spaces. See AppendixA.3 of [HZ94] for the analogous proof in the case of loops taking values in R n . Lemma 1.1.
Given any s ≥ , let us consider the continuous inclusion T : H s ( S , H ) → L ( S , H ) . The adjoint operator T ∗ can be represented as T ∗ ( y ) = y + X k =0 π | k | s e πkJt y k , (2) and if f : H s ( S , H ) → R is a differentiable map, then the following equalityholds T ∗ ∇ L f ( x ) = ∇ s f ( x ) , (3) for any x ∈ H s ( S , H ) . roof. Using the definition of adjoint we get X k ∈ Z h x k , y k i = h x , T ∗ ( y ) i + 2 π X k =0 | k | s h x k , T ∗ ( y ) k i for any x ∈ H s and y ∈ L , thus the equality (2).For any x, y ∈ H s we get h T ∗ ∇ L f ( x ) , y i s = h∇ L f ( x ) , T y i L = h∇ L f ( x ) , y i L = df ( x )[ y ] = h∇ s f ( x ) , y i s hence the equality (3).We are now ready to focus on the properties of the Hamiltonian part ofthe action functional A H , namely b H ( x ) = Z H ( x ( t )) dt, with H ∈ C ∞ ( H , R ) autonomous Hamiltonian. Proposition 1.2.
If all the derivatives of H ∈ C ∞ ( H , R ) have at most poly-nomial growth, i.e. if for every k ∈ N there are constants C k and N k suchthat k d k H ( x ) k ≤ C k (1 + k x k N k ) ∀ x ∈ H , then the functional b H : E → R is C ∞ . Moreover d k b H ( x )[ u ] k = Z d k H ( x ( t ))[ u ( t )] k dt ∀ x, u ∈ E. Proof.
The estimate on the growth of H and the fact that H ( S , H ) iscontinuously embedded in any L p space for p = ∞ imply that b H is definedon E . The Taylor expansion of H allows to recover the Taylor series of b H and again using the hypothesis on H and the continuity of the embedding of H ( S , H ) into any L p ( S , H ) one deduces that b H has the same regularityas H .Starting from this point all the Hamiltonians H ∈ C ∞ ( H , R ) we considerare assumed to have globally Lipschitz continuous gradient H → H x
7→ ∇ H ( x ) . emma 1.2. The map H ( S , H ) → H ( S , H ) x
7→ ∇ b H ( x ) is globally Lipschitz continuous.Proof. If we consider the inclusion T : H ( S , H ) → L ( S , H ) , equality (2) implies that | T ∗ ( y ) | ≤ | y | L . (4)Using (3) we obtain T ∗ ∇ H ( x ) = T ∗ ∇ L b H ( x ) = ∇ b H ( x ) and by (4) together with the assumption that ∇ H is Lipschitz continuous on H , we get that |∇ b H ( x ) − ∇ b H ( y ) | = | T ∗ ( ∇ H ( x ) − ∇ H ( y )) | ≤ |∇ H ( x ) − ∇ H ( y ) | L ≤ C | x − y | L ≤ C | x − y | and hence |∇ b H ( x ) − ∇ b H ( y ) | ≤ C | x − y | . For any k ∈ Z , let P k : E → E be the orthogonal projector onto thespace corresponding to the k -th Fourier mode ( P k x )( t ) = e πktJ x k , x k ∈ H . We search for an estimate on the norm of the k -th Fourier coefficients ∇ b H ( x ) k of ∇ b H ( x ) . 12 emma 1.3. Let H ∈ C ∞ ( H , R ) be a Hamiltonian whose second order dif-ferential is bounded. For any ≤ s ≤ there exists a constant C s > suchthat k∇ b H ( x ) k k ≤ C s (1 + | x | s ) | k | s +1 for any k ∈ Z \{ } and any x ∈ H s ( S , H ) .Moreover if y ∈ H s ( S , H ) it holds k∇ b H ( x ) k − ∇ b H ( y ) k k ≤ |∇ H ( x ) − ∇ H ( y ) | s π | k | s +1 . Proof.
For k = 0 we notice that | P k x | s = √ π | k | s k x k k . (5)We compute | P k ∇ s b H ( x ) | s = max | u | s =1 h P k ∇ s b H ( x ) , u i s = max | u | s =1 h∇ s b H ( x ) , P k u i s = max | u | s =1 db H ( x )[ P k u ] = max | u | s =1 Z h∇ H ( x )( t ) , ( P k u )( t ) i dt = max y ∈ H π | k | s k y k =1 Z h∇ H ( x )( t ) , e πktJ y i dt = max y ∈ H π | k | s k y k =1 Z h e − πktJ ∇ H ( x )( t ) , y i dt = max y ∈ H k y k = √ π | k | s h y, Z e − πktJ ∇ H ( x )( t ) dt i = max y ∈ H k y k = √ π | k | s h y, ∇ H ( x ) k i = 1 √ π | k | s k∇ H ( x ) k k | P k ∇ s b H ( x ) | s = k∇ H ( x ) k k√ π | k | s = | P k ∇ H ( x ) | s π | k | s . Using the integral definition of the H s -norm and the assumptions on theboundedness of the second differential of H we get the estimate | P k ∇ H ( x ) | s ≤ |∇ H ( x ) | s ≤ |∇ H ( x ) + ∇ H (0) − ∇ H (0) | s ≤ |∇ H ( x ) − ∇ H (0) | s + |∇ H (0) | s ≤ c s (1 + | x | s ) , for any s ≤ .In conclusion, combining (5) with the two estimates above we get k∇ s b H ( x ) k k = | P k ∇ s b H ( x ) | s π | k | s ≤ C s (1 + | x | s ) | k | s . (6)For any ≤ s ≤ and any integer k = 0 , it follows from Lemma 1.1 that k∇ s b H ( x ) k k = k∇ b H ( x ) k k| k | s − hence using (6) we deduce k∇ b H ( x ) k k ≤ C s (1 + | x | s ) | k | s +1 . Performing an analogous computation we also get that if x, y ∈ H s ( S , H ) then k∇ b H ( x ) k − ∇ b H ( y ) k k ≤ |∇ H ( x ) − ∇ H ( y ) | s π | k | s +1 . In the case H = R n , under the assumption on the polynomial growth ofthe derivatives of H , it is possible to deduce that the map E → Ex
7→ ∇ b H ( x )
14s compact, namely that sends bounded sets into precompact sets (i.e. setswhose closure is compact). This follows from the Sobolev’s embedding the-orem which implies that H ( S , R n ) is compactly embedded in L ( S , R n ) .Nevertheless the following simple example, which serves as a prototype forthe situation we will encounter in a more general case, shows that the map x
7→ ∇ b H ( x ) is in general not compact if H is infinite dimensional. Example 1.1.
Let us consider the quadratic time-independent Hamiltonian H : H → R defined as H ( y ) = µ k y k , with µ ∈ R . We have that b H ( x ) = Z µ k x ( t ) k dt = µ h x, x i L . By representing x as a Fourier series we get h x, x i L = X k ∈ Z h x k , x k i , and h x, x i = h x , x i + 2 π X k =0 | k |h x k , x k i . The k -th Fourier coefficient of ∇ L b H ( x ) = ∇ H ( x ) is ∇ H ( x ) k = 2 µx k , while the Fourier coefficients of ∇ b H ( x ) are ∇ b H ( x ) = 2 µx , ∇ b H ( x ) k = 2 µ π | k | x k for k = 0 , thus the map x
7→ ∇ b H ( x ) is not compact as long as µ = 0 because, forexample, it sends any bounded non precompact set of constant curves into anon precompact sets of curves of E .We conclude the section by reviewing some general and well known resultsin critical point theory.Let H be a separable Hilbert space and f ∈ C ( H , R ) .15 efinition 1.1. A Palais-Smale sequence for f at level c ∈ R , abbreviatedas (PS) c , is a sequence { x n } n ∈ N of elements of H such that ∇ f ( x n ) −→ n → + ∞ in H and f ( x n ) −→ n → + ∞ c . The map f satisfies the Palais-Smale condition atlevel c if any (PS) c sequence of f admits a convergent subsequence.Let F be a family of subsets of H , we define the minimax value c ( f, F ) belonging to f and F as c ( f, F ) := inf F ∈F sup x ∈ F f ( x ) ∈ R ∪ {−∞} ∪ { + ∞} . Theorem 1.1 (Minimax lemma) . Assume f and F meet the following con-ditions:1) ˙ x = −∇ f ( x ) defines a global flow ϕ t ( x ) ,2) the family F is positively invariant under the flow, i.e., if F ∈ F then ϕ t ( F ) ∈ F for every t ≥ ,3) −∞ < c ( f, F ) < ∞ ,then there exists a (PS) sequence for f at level c ( f, F ) . If in addition4) f satisfies (PS) at level c ( f, F ) ,then f has a critical point with critical value c ( f, F ) .Proof. See [HZ94], Section 3.2.
In this section we study the properties of (PS) c sequences of the Hamil-tonian action functional A H : E → R under the assumption that the map H → H x
7→ ∇ H ( x )
16s globally Lipschitz continuous. The first result we obtain is that sym-plectomorphisms satisfying reasonable boundedness conditions induce a cor-respondence between bounded (PS) c sequences of the Hamiltonian actionfunctional. Proposition 2.1.
Let ϕ : H → H be a symplectic diffeomorphism such that ϕ and ϕ − have bounded differentials up to the second order, let H ∈ C ∞ ( H , R ) be a Hamiltonian whose gradient is globally Lipschitz continuous and considerthe Hamiltonian G = H ◦ ϕ − .Let s > be a real number, if { x n } n ∈ N ∈ E is a H s -bounded (PS) c sequencefor A H then { ϕ ∗ ( x n ) } n ∈ N is a (PS) c sequence for A G . We remark that the assumption on the H s -boundedness of (PS) c se-quences with s > is not really restrictive, indeed in the final part of thesection we will see that in our setting any H -bounded (PS) c sequence canbe modified into a H s -bounded (PS) c sequence for any s < .Let ϕ : H → H be any symplectomorphism and consider the compositionoperator (known in the literature, in a more general context, as superposi-tion operator) defined as ϕ ∗ : E → Im ( ϕ ∗ ) u ϕ ◦ u. The differential of ϕ ∗ at a point x ∈ E , if it exists, is given by Ä Dϕ ∗ ( x )[ u ] ä ( t ) = Ä dds (cid:12)(cid:12)(cid:12) ϕ ∗ ( x + su ) ä ( t ) = Ä dds (cid:12)(cid:12)(cid:12) ϕ ( x ( t ) + su ( t )) ä = Dϕ ( x ( t ))[ u ( t )] , ∀ u ∈ E. We define the formal differential of ϕ ∗ as the multiplication operator Dϕ ∗ ( x )[ u ] = Dϕ ( x )[ u ] . Lemma 2.1.
Let ϕ : H → H be a diffeomorphism with bounded differentialsup to the second order, then ϕ ∗ : H ( S , H ) → H ( S , H ) s a continuous map.Moreover for any s > there is a constant C s such that for every x ∈ H s ( S , H ) the operator Dϕ ∗ ( x ) is bounded on H ( S , H ) , i.e. | Dϕ ∗ ( x ) | L ( E,E ) ≤ C s ( | x | H s ( S , H ) + 1) . Proof.
The map ϕ is globally Lipschitz thus ϕ ∗ : L ( S , H ) → L ( S , H ) isglobally Lipschitz; moreover it is possible to find a constant K such that | ϕ ( x ) | sem, = Z Z | ϕ ( x ( t )) − ϕ ( x ( r )) | | t − r | dtdr ≤ Z Z K | x ( t ) − x ( r ) | | t − r | dtdr = K Z Z | x ( t ) − x ( r ) | | t − r | dtdr = K | x | sem, hence ϕ ∗ : E → E is well defined.Our next step is to prove the continuity of ϕ ∗ : H ( D , H ) → H ( D , H ) , with D disk whose boundary is S .The diffeomorphism ϕ is C and has bounded differential, thus the chainrule for Sobolev spaces W ,p ( D , H ) (which can be deduced using the standardchain rule for smooth maps and the dominated convergence theorem) impliesthat if u ∈ H ( D , H ) then ϕ ∗ ( u ) ∈ H ( D , H ) ∇ ( ϕ ∗ ( u )) = Dϕ ∗ ( u )[ ∇ u ] . Let us consider a sequence { u n } n ∈ N ∈ H ( D , H ) such that u n −→ n → + ∞ u in H ( D , H ) . Since u n −→ n → + ∞ u in H ( D , H ) , up to the choice of a subsequencewe have pointwise convergence almost everywhere, namely u n k ( x ) −→ n k → + ∞ u ( x ) ∀ x ∈ D \ Σ ∇ u n k ( x ) −→ n k → + ∞ ∇ u ( x ) ∀ x ∈ D \ Σ Σ null-subset of D .By the Lipschitz continuity of ϕ and the dominated convergence theorem itfollows that ϕ ∗ ( u n ) −→ n → + ∞ ϕ ∗ ( u ) in L ( D , H ) . Using the continuity and boundedness assumption on the differential of ϕ wededuce that Dϕ ( u n k ( x ))[ ∇ u n k ( x )] −→ n k → + ∞ Dϕ ( u ( x ))[ ∇ u ( x )] ∀ x ∈ D \ Σ . By the dominate convergence theorem we get Dϕ ∗ ( u n k )[ ∇ u n k ] −→ n k → + ∞ Dϕ ∗ ( u )[ ∇ u ] in L ( D , H ) , and using chain rule it follows ∇ ( ϕ ∗ ( u n k )) −→ n k → + ∞ ∇ ( ϕ ∗ ( u )) in L ( D , H ) , hence the desired continuity of ϕ ∗ in H ( D , H ) .To recover the continuity of ϕ ∗ on H ( S , H ) we need to do a small digression.It is known that given any element of H ( S , H ) there is a unique extensionof u to an element ˜ u of H ( D , H ) such that ∂ ˜ u = u and ∆˜ u = 0 in the interiorof D . The map R : H ( S , H ) → H ( D , H ) u ˜ u is an isometry called harmonic extension and it is a right inverse of the traceoperator ∂ : H ( D , H ) → H ( S , H ) u u | ∂ D . Both these linear maps are continuous.Let us take a converging sequence u n −→ n → + ∞ u in H ( S , H ) .It follows that | ˜ u n − ˜ u | H ( D ) = | R ( u n − u ) | H ( D ) ≤ C | u n − u | H ( S ) −→ n → + ∞ , ϕ ∗ : H ( D , H ) → H ( D , H ) imply | ϕ ∗ ( u n ) − ϕ ∗ ( u ) | H ( S ) = | ∂ Ä ϕ ∗ (˜ u n ) − ϕ ∗ (˜ u ) ä | H ( S ) ≤ C | ϕ ∗ (˜ u n ) − ϕ ∗ (˜ u ) | H ( D ) −→ n → + ∞ , hence the continuity of ϕ ∗ : E → E .Our next goal is to prove the existence of the differential of ϕ ∗ : H ( S , H ) → H ( S , H ) at points belonging to H s ( S , H ) with s > .Using the fact that the harmonic extension of a loop u minimizes the Dirichletintegral among all functions with trace u , together with the boundednessassumptions on Dϕ and on D ϕ we compute | Dϕ ∗ ( u )[ v ] | H ( S ) ≤ C | RDϕ ∗ ( u )[ v ] | H ( D ) ≤ C | Dϕ ∗ (˜ u )[˜ v ] | H ( D ) = C ( | Dϕ ∗ (˜ u )[˜ v ] | L ( D ) + | D ( Dϕ ∗ (˜ u )[˜ v ]) | L ( D ) ) ≤ C ( C | ˜ v | L ( D ) + | D ϕ (˜ u )[ ∇ ˜ u, ˜ v ] | L ( D ) + | Dϕ (˜ u )[ ∇ ˜ v ] | L ( D ) ) ≤ C ( C | ˜ v | L ( D ) + C |∇ ˜ u | L p ( D ) | ˜ v | L q ( D ) + C |∇ ˜ v | L ( D ) ) where by the Hölder’s inequality the last estimate holds for p > arbitraryand q equal to twice the conjugate exponent of p .Since | ˜ v | L ( D ) , | ˜ v | L q ( D ) and |∇ ˜ v | L ( D ) can be bounded by the quantity | ˜ v | H ( D ) ,which is bounded by the value | v | H ( S ) multiplied with a constant, we inferthat | Dϕ ∗ ( u )[ v ] | H ( S ) ≤ C ( |∇ ˜ u | L p ( D ) + 1) | v | H ( S ) . (7)If we choose t ∈ (1 , and we take p = 1 + t − t > , then the Sobolev’s em-bedding theorem implies that H t ( D , H ) embeds continuously into W ,p ( D , H ) ,therefore | Dϕ ∗ ( u )[ v ] | H ( S ) ≤ C t ( |∇ ˜ u | H t ( D ) + 1) | v | H ( S ) ≤ C t ( |∇ u | H t − ( S ) + 1) | v | H ( S ) , where the last inequality follows since the harmonic extension is a boundedright inverse of the trace operator ∂ : H t ( D , H ) → H t − ( S , H ) .20e are now ready to prove Proposition 2.1. Proof.
The elements of H ( S , H ) are absolutely continuous loops, thus theiraction is preserved by symplectomorphisms between simply connected do-mains. Indeed if x is an absolutely continuous loop then − Z h J ( ϕ ◦ x ) ′ ( t ) , ϕ ( x ( t )) i dt = Z S x ∗ ( ϕ ∗ λ ) = Z S x ∗ ( λ + dh ) = Z S x ∗ ( λ )= − Z h J x ′ ( t ) , x ( t ) i dt, where ϕ ∗ λ − λ is exact and therefore it is the differential of a smooth function h . The Hamiltonian part of the action functional does not change, since bydefinition Z G ( ϕ ( x ( t ))) dt = Z H ( x ( t )) dt, thus we conclude that A G ( ϕ ∗ ( x )) = A H ( x ) . (8)The density of H ( S , H ) ֒ → H ( S , H ) together with the continuity of themap ϕ ∗ : E → E and of A H imply that the equality (8) holds for any loop x ∈ E . In particular, given any (PS) c sequence { x n } n ∈ N ∈ E we have A G ( ϕ ∗ ( x n )) = A H ( x n ) , for any n . Let { x n } n ∈ N be a H s -bounded (PS) c sequence with s > ; if wedifferentiate the equation above, in view of Lemma 2.1 we can write d A G ( ϕ ∗ ( x n )) = d A H ( x n ) Dϕ − ∗ ( ϕ ∗ ( x n )) , thus ∇ A G ( ϕ ∗ ( x n )) = Dϕ − T ∗ ( ϕ ∗ ( x n )) ∇ A H ( x n ) . Lemma 2.1 also implies the boundedness of Dϕ − ∗ : x
7→ L ( E, E ) , c such that |∇ A G ( ϕ ∗ ( x n )) | ≤ | Dϕ − T ∗ ( ϕ ∗ ( x n )) | L ( E,E ) |∇ A H ( x n ) | ≤ c |∇ A H ( x n ) | . This, combined with equality (8) imply that if { x n } n ∈ N is an H s -bounded(PS) c sequence of A H , then { ϕ ∗ ( x n ) } n ∈ N is a (PS) c sequence of A G .The next two results serve to approximate bounded (PS) sequences withequivalent (PS) sequences fulfilling stricter boundedness conditions (we saythat two (PS) c sequences { x n } n ∈ N and { y n } n ∈ N are equivalent if k x n − y n k −→ n → + ∞ and they share the same almost critical level c ). Lemma 2.2.
Let H ∈ C ∞ ( H , R ) be a Hamiltonian whose second order dif-ferential is bounded. Let ≤ r ≤ and { x n } n ∈ N be a H r -bounded (PS) c sequence of A H : E → R such that ∇ A H ( x n ) is H r -infinitesimal. Then theelements y n := x n − ( P + − P − ) ∇ A H ( x n ) define a H s -bounded (PS) c sequence { y n } n ∈ N for any s < r + , such that | y n − x n | r = o (1) . Moreover the following estimate holds |∇ A H ( y n ) | s ≤ o (1) + c s |∇ H ( x n ) − ∇ H ( y n ) | r . Proof.
Let
R > be an upper bound for | x n | r . By assumption the sequence ∇ A H ( x n ) = ( P + − P − ) x n − ∇ b H ( x n ) (9)is infinitesimal in H r ( S , H ) .The sequence given by y n := x n − ( P + − P − ) ∇ A H ( x n )
22s (PS) c because A H and its gradient are uniformly continuous on boundedsets.For any k ∈ Z , if we apply the projector P k to (9) we get P k ∇ A H ( x n ) = ( sgn k ) P k x n − P k ∇ b H ( x n ) , thus P k y n = P k x n − ( sgn k ) P k ∇ A H ( x n ) = ( sgn k ) P k ∇ b H ( x n ) , hence y kn = ( sgn k ) ∇ b H ( x n ) k . By Lemma 1.3, for any k = 0 we have the estimate k y kn k ≤ c ( R + 1) | k | r +1 , therefore | y n | H s = k y n k + 2 π X k =0 | k | s k y kn k ≤ R + 2 πc ( R + 1) X k =0 | k | r − s +1) where the series above converges for any s < r + , thus we get the desiredbound.To prove the second part of the statement we rewrite the H -gradient of A H as ∇ A H ( y n ) = ( P + − P − )( x n − ( P + − P − ) ∇ A H ( x n )) − ∇ b H ( y n )= ( P + − P − ) x n − ( P + + P − ) ∇ A H ( x n ) − ∇ b H ( y n )= ( P + − P − ) x n − ∇ A H ( x n ) + P ∇ A H ( x n ) − ∇ b H ( y n )= ( P + − P − ) x n − ( P + − P − ) x n + ∇ b H ( x n ) + P ∇ A H ( x n ) − ∇ b H ( y n )= P ∇ A H ( x n ) + ∇ b H ( x n ) − ∇ b H ( y n ) . |∇ A H ( y n ) | s ≤ | P ∇ A H ( x n ) | s + 2 π X k =0 | k | s k∇ b H ( x n ) k − ∇ b H ( y n ) k k + k∇ b H ( x n ) − ∇ b H ( y n ) k ≤ k P ∇ A H ( x n ) k + c |∇ H ( x n ) − ∇ H ( y n ) | r X k =0 | k | r − s +1) . The first quantity in the last line is infinitesimal since { x n } n ∈ N is (PS), andthe series appearing in the second quantity converges for any s < r + , thuswe get the desired estimate. Lemma 2.3.
Let H ∈ C ∞ ( H , R ) be a Hamiltonian whose second and thirdorder differentials are bounded. Let { x n } n ∈ N be a bounded (PS) c sequenceof A H , then for any ≤ s < we can find an equivalent (PS) c sequence { y n } n ∈ N which is H s -bounded and such that ∇ A H ( y n ) is infinitesimal in the H s -norm.Proof. We start by applying the first part of Lemma 2.2 to obtain, for s < ,a H s -bounded (PS) c sequence { z n } n ∈ N which is equivalent to { x n } n ∈ N andthen we modify this new sequence to get an equivalent one u n := z n − ( P + − P − ) ∇ A H ( z n ) . By Lemma 2.2 the sequence { u n } n ∈ N is H s -bounded and ∇ A H ( u n ) is H s -infinitesimal for any s < , where the latter claim follows by using the meanvalue theorem and the fact that the composition operator H ( S , H ) → H ( S , H ) u
7→ ∇ H ( u ) has bounded differential if restricted to elements of H r ( S , H ) with r > (for this we have to require the boundedness of the second and third orderdifferentials of H ; see the analogous proof we gave for the operator ϕ ∗ ).Now we introduce another equivalent sequence v n := u n − ( P + − P − ) ∇ A H ( u n ) , H s + -norm for any s < because of the first partof Lemma 2.2, and for which ∇ A H ( v n ) is H s -infinitesimal.Finally we define another equivalent sequence y n := v n − ( P + − P − ) ∇ A H ( v n ) . The composition operator H ( S , H ) → H ( S , H ) u
7→ ∇ H ( u ) has bounded differential if restricted to elements of H t ( S , H ) with t > ,therefore by interpolation with the composition operator defined on H ( S , H ) we get that for any ≤ q ≤ , the operator H q ( S , H ) → H q ( S , H ) u
7→ ∇ H ( u ) has bounded differential if restricted to elements of H t ( S , H ) , indeed by theinterpolation formula we have | D ∇ H ( u ) | L ( H q ,H q ) ≤ | D ∇ H ( u ) | θ L ( H ,H ) | D ∇ H ( u ) | − θ L ( H ,H ) , for any q = 1 − θ with θ ∈ (0 , . As a corollary of Lemma 2.2 we deducethat the sequence ∇ A H ( y n ) is H s -infinitesimal for any s < .As an aside remark we observe that (PS) c sequence { y n } n ∈ N appearing inthe statement of the proposition above can be made H s -bounded for any s > , if additional conditions on the boundedness of the higher order differentialsof H are fulfilled.The L -gradient of A H is readily computed as ∇ L A H ( x ) = − Ä J ˙ x + ∇ H ( x ) ä ,
25n fact d A H ( x )[ u ] = dds (cid:12)(cid:12)(cid:12) s =0 A H ( x + su )= − Z h J ˙ u ( t ) , x ( t ) i dt − Z h J ˙ x ( t ) , u ( t ) i dt − Z dH ( x ( t ))[ u ( t )] dt = − Z h J ˙ x ( t ) + ∇ H ( x ( t )) , u ( t ) i dt. If { x n } n ∈ N is a (PS) c sequence found by means of the proposition above weget that ∇ L A H ( x n ) is infinitesimal in the L -norm, indeed using Lemma 1.1we deduce |∇ A H ( x n ) | ≤ |∇ L A H ( x n ) | L ≤ π |∇ A H ( x n ) | , thus the loop z n ( t ) := J ˙ x n ( t ) + ∇ H ( x n ( t )) is infinitesimal in the L -norm. Proposition 2.2.
Let H ∈ C ∞ ( H , R ) be a Hamiltonian whose second andthird order differentials are bounded. Let { x n } n ∈ N be a bounded (PS) c se-quence of A H , then we can find an equivalent (PS) c sequence { y n } n ∈ N whichis H s -bounded for any s < and such that | J ˙ y n + ∇ H ( y n ) | L −→ n → + ∞ . Moreover we have that H ( y n ) −→ n → + ∞ c ∈ R ∪ {±∞} uniformly to some constant function c .Proof. The first part of the result follows from the discussion above.If we differentiate H ◦ y n with respect to t we get ddt H ( y n ( t )) = dH ( y n ( t ))[ ˙ y n ( t )] = h∇ H ( y n ( t )) , ˙ y n ( t ) i = h z n ( t ) − J ˙ y n ( t ) , ˙ y n ( t ) i = h z n ( t ) , ˙ y n ( t ) i , z n loop which is infinitesimal in the L -norm. Since { ˙ y n } n ∈ N is L -bounded, we deduce that ddt H ( y n ) defines an infinitesimal sequence in the L -norm and because y n is absolutely continuous the conclusion follows. Let ( H , ω ) be a symplectic Hilbert space endowed with a compatible in-ner product h· , ·i and let { e i , f i } i ∈ N be a countable orthonormal basis suchthat any { e i , f i } spans a symplectic plane. For any n ∈ N we consider theorthogonal projections P n : H → H n x n X i =1 h x, e i i e i + h x, f i i f i , (10)onto the n -dimensional symplectic Hilbert subspace H n .We define a set of admissible symplectomorphisms Symp a ( H , ω, h· , ·i ) := ¶ ϕ ∈ Symp ( H , ω ) (cid:12)(cid:12)(cid:12) for k = ± , Dϕ k and D ϕ k are bounded ;( I − P n ) ϕ k | P n H −→ n → + ∞ uniformly on bounded sets ;[ P n , Dϕ k ( x ) ∗ ] −→ n → + ∞ in operators’ norm, uniformly in x ∈ H on bounded sets © . which is easy to check that is actually a group. Theorem (Infinite dimensional non-squeezing) . Let ( H , ω ) be a Hilbert sym-plectic space endowed with a compatible inner product h· , ·i , B r the ball centredin with radius r and Z R a cylinder whose basis lays on a symplectic planeand has symplectic area πR . Let ϕ ∈ Symp a ( H , ω, h· , ·i ) , if ϕ ( B r ) ⊂ Z R then r ≤ R .Proof ’s outline. The cylinder Z R can be written as Z R = { z ∈ H | x + y < R } . B s with s > cannot be symplectically embedded by ϕ ∈ Symp a ( H , ω, h· , ·i ) into the cylinder Z . Indeed if any admissible symplecto-morphism squeezes B rR into Z , then the symplectomorphism φ ∈ Symp a ( H , ω, h· , ·i ) defined as φ = RφR − squeezes B r into Z R and vice-versa if any φ squeezes B r into Z R then ϕ = R − ϕR squeezes B rR into Z .Our strategy to prove the above reformulation of the non-squeezing theoremis assuming the existence for r > of an admissible symplectic embedding ϕ : B r ֒ → Z and then showing that this leads to a contradiction.Since r > we can choose two real numbers m > π , δ > and a smoothmap g : [0 , + ∞ [ → R such that g ( t ) = 0 if t < δ,g ( t ) = m if t ≥ r − δ, ≤ g ′ ( t ) < π. We define a time-independent Hamiltonian F : H → R as F ( x ) := g ( | x | ) . Its Hamiltonian flow is supported in B r and Proposition 3.1 will show that theonly bounded (PS) c sequences of A F are at non positive levels c . NeverthelessProposition 3.2 will exhibit a (PS) c sequence of A F at positive level, hencethe initial assumption ϕ ( B r ) ⊂ Z leads to a contradiction.To complete the proof we need to show that Proposition 3.1 and Propo-sition 3.2 hold. Proposition 3.1.
The action functional A F : E → R associated to the time-independent Hamiltonian F ( x ) = g ( | x | ) has no bounded (PS) c sequences atany positive level c .Proof. We know that ∇ F ( x ) = 2 g ′ ( | x | ) x, g there is a real number ǫ > such that g ′ ( t ) < π − ǫ, for any t. Given any bounded (PS) c sequence of loops, according to Proposition 2.2 wecan find an equivalent H -bounded (PS) c sequence { x n } n ∈ N such that g ( | x n ( t ) | ) = F ( x n ( t )) −→ n → + ∞ c ∈ R uniformly in t , hence we deduce the uniform convergence of g ′ ( | x n ( t ) | ) −→ n →∞ d ∈ R . This implies that ∇ F ( x n ( t )) − dx n ( t ) −→ n →∞ uniformly in t , and hence ∇ F ( x n ) − dx n −→ n →∞ in L ( S , H ) . From Lemma 1.1 it follows that |∇ b F ( x n ) − dT ∗ x n | = | T ∗ ( ∇ F ( x n ) − dx n ) | ≤ |∇ F ( x n ) − dx n | L , thus ∇ b F ( x n ) − dT ∗ x n −→ n →∞ in H ( S , H ) . In order for x n = ( x − n , x n , x + n ) to define a (PS) c sequence for A F it is necessarythat ∇ a ( x n ) − ∇ b F ( x n ) −→ n →∞ in the H -norm, thus it is necessary that x + n − x − n − dT ∗ x n = ∇ a ( x n ) − dT ∗ x n −→ n →∞ in H ( S , H ) . (11)Because of Lemma 1.1 we know that dT ∗ x n = 2 dx n , dT ∗ x kn = 2 d π | k | x kn for k = 0 , d < π − ǫ implies that ǫπ | x + n | = | x + n − π − ǫ )2 π x + n | ≤ | x + n − dT ∗ x + n | , | x − n | ≤ | x − n + 2 dT ∗ x − n | . Therefore (11) is possible only if | x + n | −→ n →∞ and | x − n | −→ n →∞ ; since A F ( x n ) = 12 | x + n | − | x − n | − b F ( x n ) , and ≤ b F ( x ) ≤ m, ∀ x ∈ E, we deduce that A F ( x n ) −→ n →∞ c, with − m ≤ c ≤ .The last step to prove the non-squeezing is to show that if squeezing werepossible, then we would be able to find a (PS) c sequence at level c > forthe action functional A F . In order to do this we first introduce the conceptof approximation scheme as in [CLL97] and [Abb01]. Definition 3.1.
Let X be a separable Hilbert space. An approximationscheme with respect to a bounded linear operator L ∈ L ( X, X ) is a sequence { P n } n ∈ N of finite dimensional orthogonal projections such that1) rank ( P n ) ⊂ rank ( P m ) if n ≤ m ,2) P n −→ n → + ∞ I strongly,3) [ P n , L ] := P n L − LP n −→ n → + ∞ in the operators’ norm.An explicit example of approximation scheme for the identity map is givenby the sequence { P n } n ∈ N of orthogonal projections as in (10).30et ϕ ∈ Symp a ( H , ω, h· , ·i ) be such that ϕ ( B r ) ⊂ Z with r > , we define aHamiltonian H : H → R as H ( x ) := F ( ϕ − ( x )) if x ∈ ϕ ( B r ) ,m if x / ∈ ϕ ( B r ) . The support of the Hamiltonian H − m is the set { H < m } ( ϕ ( B r ) which has positive distance from ∂ϕ ( B r ) .Indeed Dϕ − is bounded by assumption, hence by the mean value theoremfor any points x, y ∈ B r we have k x − y k = k ϕ − ( ϕ ( x )) − ϕ − ( ϕ ( y )) k ≤ C k ϕ ( x ) − ϕ ( y ) k with C positive constant, thus k x − y k C ≤ k ϕ ( x ) − ϕ ( y ) k , and this implies that if k x − y k ≥ δ , then k ϕ ( x ) − ϕ ( y ) k ≥ δ C .Let x ∈ B r , if the distance d ( x, ∂B r ) is not larger than the value δ > appearing in the definition of the function F , then F ( x ) = m .Therefore for any point y ∈ ϕ ( B r ) such that d ( y, ∂ϕ ( B r )) ≤ δC , we get H ( y ) = m and since ϕ ( ∂B r ) = ∂ ( ϕ ( B r )) we deduce the claim, namely theexistence of a real value λ > such that { H < m } + B λ ( ϕ ( B r ) . We define a quadratic form q : H → R as q ( x ) = q + p + 1 N ∞ X i =2 ( q i + p i ) , where p i , q i are the i -th coordinates of x in the basis { e i , f i } i ∈ N and N is anatural number large enough so that { H < m } + B λ ⊂ { q < } and { H < m } + B λ ⊂ ϕ ( B r ) , (12)31or some λ > . The possibility of finding such values N and λ is a con-sequence of the discussion above about the set { H < m } , together with theobservations that ∂ϕ ( B r ) ⊂ ∂Z and that ϕ sends bounded sets into boundedsets.Let us fix a real number µ such that π < µ < min { m, π } , and choose a smooth function ρ : [0 , + ∞ [ → R satisfying ρ ( t ) = m for any t ≤ ρ ( t ) ≥ tµ for any t ≥ ρ ( t ) = tµ if t ≥ M large enough < ρ ′ ( t ) ≤ µ for any t > . We introduce the Hamiltonian K ( x ) := H ( x ) if x ∈ { q < } ρ ( q ( x )) if x ∈ { q ≥ } . For any fixed n ∈ N we define the Hamiltonian K n : H n → R as K n := K | H n . Every H n is finite dimensional, hence for fixed n we can apply a standardminimax argument in order to find a critical point of A K n at positive criticallevel.We do not reprove here this deep classical result, which is the main topicof Chapter 3 of [HZ94] and is essentially equivalent to the Gromov’s non-squeezing in R n ; nevertheless we give an outline of the proof of this fact,paying particular attention to the intermediate results we need to adapt toour setting. This also gives us the opportunity to highlight what can gowrong in trying to extend the same proof to an infinite dimensional Hilbertsymplectic space setting.Let us describe the minimax setting. For n ∈ N fixed, we consider the Hilbertspace of loops E n := H ( S , H n ) , E = H ( S , H ) . If weconsider the splitting E n = E − n ⊕ E n ⊕ E + n with respect to the Fourier coefficients, then E − n ⊂ E − , E n ⊂ E and E + n ⊂ E + . We introduce the subsets ˜Γ α := { x ∈ E + (cid:12)(cid:12)(cid:12) | x | = α } ⊂ E + , ˜Γ nα := { x ∈ E + n (cid:12)(cid:12)(cid:12) | x | = α } ⊂ E + n , and we translate them by ϕ (0) to obtain Γ α := ˜Γ α + ϕ (0) , Γ nα := ˜Γ nα + ϕ (0) . The next statement is a consequence of the fact that K vanishes in a neigh-bourhood of ϕ (0) . Lemma 3.1.
For α > small enough we have that inf x ∈ Γ nα A K n ( x ) ≥ inf x ∈ Γ α A K ( x ) > . Proof.
Since E + n ⊂ E + the left inequality is trivially true; moreover K van-ishes identically in a neighbourhood of ϕ (0) , hence Proposition 1.2 yieldsto A K ( ϕ (0)) = 0 , d A K ( ϕ (0)) = 0 , d A K ( ϕ (0))[ u, u ] = | P + u | − | P − u | , for any u ∈ E , thus the claim follows by the Taylor formula with Peano’sreminder.We define the subsets ˜Σ τ ⊂ E and ˜Σ nτ ⊂ E n as ˜Σ τ := { x ∈ E (cid:12)(cid:12)(cid:12) x = x − + x + se + , | x − + x | ≤ τ and ≤ s ≤ τ } , ˜Σ nτ := { x ∈ E n (cid:12)(cid:12)(cid:12) x = x − + x + se + , | x − + x | ≤ τ and ≤ s ≤ τ } , τ > and e + ( t ) := e πtJ e √ π is a circle in E +1 ⊂ E + . We have | e + | = 1 , | e + | L = 12 π and we denote with ∂ ˜Σ τ (resp. with ∂ ˜Σ nτ ) the boundary of Σ τ in E − × E × R e + (resp. of Σ nτ in E − n × E n × R e + ). We denote the shift of ˜Σ τ and ˜Σ nτ by ϕ (0) with Σ τ := ˜Σ τ + ϕ (0) , Σ nτ := ˜Σ nτ + ϕ (0) . Lemma 3.2. If τ is big enough then A K (cid:12)(cid:12)(cid:12) ∂ Σ τ ≤ and A K n (cid:12)(cid:12)(cid:12) ∂ Σ nτ ≤ .Proof. To prove this we use the asymptotic behaviour of K . Since a (cid:12)(cid:12)(cid:12) E − × E and b K are non positive, it follows that A K (cid:12)(cid:12)(cid:12) E − × E ≤ .Let us assume that s = τ or that ≤ s ≤ τ and | x − ϕ (0) + x − | = τ . Since K is the quadratic form µq outside of a bounded set, we can find a constant c ≥ such that K ( x ) ≥ µq ( x ) − c, ∀ x ∈ H . The splitting E − × E × R e + is orthogonal with respect to the scalar productdefined by the quadratic form x → R q ( x ( t )) dt , thus A K ( x ) = 12 s − | x − | − Z K ( x ( t )) dt ≤ s − | x − | − µ Z q ( x ( t )) dt + c = 12 s − | x − | − µ Z q ( se + ) dt − µ Z q ( x − ϕ (0)) dt − µ Z q ( x − ) dt + c = 12 s − | x − | − µs π − µ Z q ( x − ϕ (0)) dt − µ Z q ( x − ) dt + c ≤ −
12 ( µπ − s − | x − | − µ Z q ( x − ϕ (0)) dt − µ Z q ( x − ) dt + c. Using the positivity of the quadratic form q and the inequality π < µ wededuce that if τ is large enough then the quantity above is non positive aslong as s = τ or | x − ϕ (0) + x − | = τ . In particular A K n (cid:12)(cid:12)(cid:12) ∂ Σ nτ ≤ , because ∂ Σ nτ ⊂ Σ τ . 34hroughout the entire chapter we are assuming that Dϕ is bounded,hence we can use Lemma 1.2 to deduce that the gradient equation ˙ x = −∇ A K ( x ) , x ∈ E (13)is globally Lipschitz continuous, thus it defines a unique global flow R × E → E ( t, x ) ϕ t ( x ) =: x · t, which, as well as its inverse, maps bounded sets into bounded sets.The same is clearly true also for the flow of ˙ x = −∇ A K n ( x ) = x − − x + + ∇ b K n ( x ) , x ∈ E n . (14)The vector field (14) is a compact perturbation of a linear one, indeed thecompactness of E n → E n x
7→ ∇ b K n ( x ) = T ∗ ∇ K n ( x ) is a consequence of Sobolev’s compact embedding theorem which affirmsthat the linear embedding T : H ( S , H n ) ֒ → L ( S , H n ) is a compact map.Using the variation of constants method it is not hard to obtain the followingrepresentation formula for the flow of (14). Lemma 3.3. [HZ94] (Chapter 3, Lemma 7) The flow of ˙ x = −∇A K n ( x ) admits the representation x · t = e t x − + x + e − t x + + l ( t, x ) where l : R × E n → E n is a compact map. Remark . The flow of (13) cannot be seen as a compact perturbation ofa linear flow because the Sobolev’s embedding T : H ( S , H ) ֒ → L ( S , H ) is not compact when H is infinite dimensional (cfr. Example 1.1).35aving a representation formula for the flow of (14), the main ingredientto prove the linking lemma below is the Leray-Schauder degree, which isdefined for maps which are compact perturbations of the identity. Lemma 3.4. [HZ94] (Chapter 3, Lemma 10) Let ϕ t be the flow of ˙ x = −∇ A K n ( x ) , then for α > small enough and τ > big enough we have ϕ t (Σ nτ ) ∩ Γ nα = ∅ for any t ≥ . Remark . By the previous remark, if H is infinite dimensional the gradientflow has no compactness property and this prevents the possibility of provinga linking lemma by means of the Leray-Schauder degree.We finally have all the ingredients to conclude the following. Lemma 3.5.
For any fixed n ∈ N there exists a critical point x n of A K n : E n → R such that x n ( t ) ∈ ϕ ( B r ) ∩ H n for any t ∈ [0 , and < inf x ∈ Γ α A K ( x n ) ≤ A K n ( x n ) ≤ sup x ∈ Σ τ A K ( x n ) < + ∞ . Proof.
For any fixed n ∈ N the equation (14) defines a global flow ϕ t on E n and the family of sets F n := { ϕ t (Σ nτ ) (cid:12)(cid:12)(cid:12) t ≥ } , is clearly positively invariant under the flow ϕ t .The value c ( A K n , F n ) := inf t ≥ sup x ∈ ϕ t (Σ nτ ) A K n ( x ) is finite, indeed A K n ( ϕ t ( x )) is a decreasing function of t and A K is boundedon bounded sets, thus inf t ≥ sup x ∈ ϕ t (Σ nτ ) A K n ( x ) ≤ sup x ∈ Σ nτ A K n ( x ) ≤ sup x ∈ Σ τ A K ( x ) < + ∞ . Moreover Lemma 3.4 implies that if τ is big enough then sup x ∈ ϕ t (Σ nτ ) A K n ( x ) ≥ inf x ∈ Γ nα A K n ( x ) ≥ inf x ∈ Γ α A K ( x ) > t ≥ , hence c ( A K n , F n ) = inf t ≥ sup x ∈ ϕ t (Σ nτ ) A K n ( x ) > . We can therefore apply the minimax lemma (Theorem 1.1) and, for anyfixed n , we get a (PS) c sequence { x kn } k ∈ N ∈ H ( S , H n ) . Since H n is finitedimensional, one can show that the functional A K n satisfy the (PS) condition(see Lemma 6, Chapter 3 of [HZ94]), therefore for any fixed n we can find acritical point x n whose critical level is positive. Moreover it is not hard tosee that any x n is supported in ϕ ( B r ) ∩ H n (see Proposition 2, Chapter 3 of[HZ94]).A critical point of A K n : E n → R at level c n is a solution x n of ˙ x n = J ∇ K n ( x n ) A K n ( x n ) = c n , hence if we apply Lemma 3.5 we can find a sequence { x n } n ∈ N of loops sup-ported in ϕ ( B r ) such that ˙ x n = J ∇ K n ( x n )0 < δ ≤ A K n ( x n ) ≤ ∆ < + ∞ . (15)It is now time to use the properties of approximation schemes in order to showthat the existence of a sequence { x n } n ∈ N as in (15) implies the existence ofa (PS) c sequence for A F at positive level. Proposition 3.2.
Let ( H , ω ) be a symplectic Hilbert space endowed witha compatible inner product and { P n } n ∈ N the approximation scheme of or-thogonal projections onto finite dimensional symplectic subspaces. Let ϕ ∈ Symp ( H , ω ) be a symplectic diffeomorphism such that Dϕ , Dϕ − are boundedandi) ( I − P n ) ϕ − | P n H −→ n → + ∞ uniformly on bounded sets, i) [ P n , Dϕ − ( x ) ∗ ] −→ n → + ∞ in operators’ norm, uniformly in x ∈ H onbounded sets.If { y n } n ∈ N is a sequence of critical points for A K n : E n → R as in (15) , thenthe sequence { x n } n ∈ N of loops x n := ϕ − ( y n ) admits a (PS) c subsequence atpositive level for the Hamiltonian action functional A F : E → R .Proof. Let us set φ := ϕ − . For any n ∈ N , if we consider the inclusion i n : P n H → H and the Hamiltonian K n := K ◦ i n , we get ∇ K n ( z ) = ( Di n ) ∗ ∇ K ( z ) = P n ∇ K ( z ) , (16)for any z ∈ H . Therefore for any critical point y n of A K n it holds ˙ y n ( t ) = J ∇ K n ( y n ( t )) = J P n ∇ K ( y n ( t )) . (17)We recall that K | ϕ ( B r ) = F ◦ φ : ϕ ( B r ) → R with F radial Hamiltonian and that the loops y n are supported in ϕ ( B r ) ∩ P n H . Using (17) together with the fact that φ is symplectic, for the loops x n = φ ( y n ) we compute ˙ x n ( t ) = Dφ ( y n ( t )) ˙ y n ( t ) = Dφ ( y n ( t )) J P n ∇ K ( y n ( t ))= J Dφ ( y n ( t )) −∗ P n Dφ ( y n ( t )) ∗ ∇ F ( x n ( t ))= J Dφ ( y n ( t )) −∗ Dφ ( y n ( t )) ∗ P n ∇ F ( x n ( t ))++ J Dφ ( y n ( t )) −∗ [ P n , Dφ ( y n ( t )) ∗ ] ∇ F ( x n ( t ))= J P n ∇ F ( x n ( t )) + J Dφ ( y n ( t )) −∗ [ P n , Dφ ( y n ( t )) ∗ ] ∇ F ( x n ( t )) . (18)The gradient of F : H → R writes as ∇ F ( z ) = 2 g ′ ( k z k ) z, thus we get ∇ F ( x n ( t )) = 2 g ′ ( k x n ( t ) k ) x n ( t ) = 2 g ′ ( k x n ( t ) k ) φ ( y n ( t )) P n ∇ F ( x n ( t )) = ∇ F ( x n ( t )) − ( I − P n )2 g ′ ( k x n ( t ) k ) φ ( y n ( t )) , (19)for any t ∈ [0 , .By substituting (19) we can rewrite (18) as ˙ x n ( t ) = J Ä ∇ F ( x n ( t )) − ( I − P n )2 g ′ ( k x n ( t ) k ) φ ( y n ( t )) ä −− J Dφ ( y n ( t )) −∗ [ P n , Dφ ( y n ( t )) ∗ ] ∇ F ( x n ( t )) , which, since ∇ L A F ( x n ) = − Ä J ˙ x n + ∇ F ( x n ) ä , is equivalent to −∇ L A F ( x n ) = ( I − P n )2 g ′ ( k x n ( t ) k ) φ ( y n ( t )) + Dφ ( y n ( t )) −∗ [ P n , Dφ ( y n ( t )) ∗ ] ∇ F ( x n ( t )) . Since y n ( t ) ∈ ϕ ( B r ) ∩ P n H , using assumption i) we get ( I − P n )2 g ′ ( k x n ( t ) k ) φ ( y n ( t )) −→ n → + ∞ uniformly in t, moreover, by assumption ii) for any t ∈ [0 , we have [ P n , Dφ ( y n ( t )) ∗ ] −→ n → + ∞ in the operators’ norm . Thus applying the dominated convergence theorem we deduce ∇ L A F ( x n ) −→ n → + ∞ in L ( S , H ) . Finally, because of the inequality |∇ A F ( x n ) | = | T ∗ ∇ L A F ( x n ) | ≤ |∇ L A F ( x n ) | L , we get ∇ A F ( x n ) −→ n → + ∞ in H ( S , H ) .
39o find the sought (PS) sequence for A F we need a small last step.By construction of the sequence { y n } n ∈ N (cfr. (15)), we have that < δ ≤ A K n ( y n ) ≤ ∆ < + ∞ . with δ, ∆ ∈ R independent on n .The symplectic action of closed orbits is preserved under symplectomor-phisms, hence < δ ≤ A F ( x n ) = A K n ( y n ) ≤ ∆ < + ∞ . Thus we can find a subsequence { x n k } n k ∈ N such that A F ( x n k ) −→ n → + ∞ c > ,namely the (PS) c sequence we were looking for.The non-squeezing theorem we just proved generalizes the one obtainedin [Kuk95a] which applies to the family of so called elementary symplecto-morphisms. We do not give the rather technical definition of elementarysymplectomorphism, but we remark that any such symplectomorphism is acompact perturbation of a linear map, which satisfies the following proper-ties. Lemma 3.6. [Kuk95a] (Lemma 3) Let ( H , ω ) be a Hilbert symplectic spaceendowed with a compatible inner product h· , ·i , B r the ball centred in withradius r and ϕ : B r → H be an elementary symplectomorphism. Then forany ǫ > and r < + ∞ there exists a natural number n such that ϕ ( x ) = L ( I + ϕ ǫ )( I + ϕ n )( x ) (20)for every x ∈ B r , where L : H → H is a direct sum of rotations in thesymplectic planes spanned by { e i , f i } , while I + ϕ ǫ : H → H and I + ϕ n : H → H are smooth symplectomorphisms such that k ϕ ǫ ( y ) k ≤ ǫ for any y ∈ ( I + ϕ n ) B r , and ϕ n ( P n x, ( I − P n ) x ) = ( ϕ n ( P n x ) , ( I − P n ) x ) . emma 3.7. [Kuk95a] (Lemma 6) Let U ⊂ H be a bounded open set and ϕ : U → H be an elementary symplectomorphism such that ϕ − : ϕ ( U ) → H is bounded, then ϕ − is also an elementary symplectomorphism.Using the two lemmata above and the infinite dimensional non-squeezingtheorem, we deduce the following. Corollary 3.1 (Kuksin’s infinite dimensional non-squeezing) . Let ( H , ω ) bea Hilbert symplectic space endowed with a compatible inner product h· , ·i , B r the ball centred in with radius r and Z R a cylinder whose basis lays ona symplectic plane and has symplectic area πR . Let ϕ : B r → H be anelementary symplectomorphism such that the differentials of ϕ and ϕ − arebounded up to the second order; if ϕ ( B r ) ⊂ Z R then r ≤ R .Proof. By assumption ϕ is an elementary symplectomorphism, hence it canbe represented as a rotation composed with a small perturbation of a finitedimensional map as in (20). Since P n commutes with I , L and ϕ n we obtain ( I − P n ) ϕ | P n H −→ n → + ∞ uniformly on bounded sets . By Lemma 3.7 we know that also ϕ − is an elementary symplectomorphism,therefore by an analogous argument we obtain ( I − P n ) ϕ − | P n H −→ n → + ∞ uniformly on bounded sets . We compute Dϕ ( x ) = LD Ä ( I + ϕ ǫ )( I + ϕ n )( x ) ä = L ( I + Dϕ ǫ ( y ))( I + Dϕ n ( x )) where y = ( I + ϕ n )( x ) , thus Dϕ ( x ) ∗ = ( I + Dϕ n ( x ) ∗ )( I + Dϕ ǫ ( y ) ∗ ) L ∗ . This implies P n Dϕ ( x ) ∗ = P n ( I + Dϕ n ( x ) ∗ )( I + Dϕ ǫ ( y ) ∗ ) L ∗ = P n L ∗ + P n Dϕ n ( x ) ∗ L ∗ + P n Dϕ ǫ ( y ) ∗ L ∗ + P n Dϕ n ( x ) ∗ Dϕ ǫ ( y ) ∗ L ∗ Dϕ ( x ) ∗ P n = ( I + Dϕ n ( x ) ∗ )( I + Dϕ ǫ ( y ) ∗ ) L ∗ P n = L ∗ P n + Dϕ n ( x ) ∗ L ∗ P n + Dϕ ǫ ( y ) ∗ L ∗ P n + Dϕ n ( x ) ∗ Dϕ ǫ ( y ) ∗ L ∗ P n . Since [ P n , L ∗ ] = [ P n , L − ] = 0 and [ P n , Dϕ n ( x ) ∗ L ∗ ] = 0 we get [ P n , Dϕ ( x ) ∗ ] = [ P n , Dϕ ǫ ( y ) ∗ L ∗ ] + [ P n , Dϕ n ( x ) ∗ Dϕ ǫ ( y ) ∗ L ∗ ] . (21)Since D ϕ is bounded, using the mean value theorem we deduce that k Dϕ ǫ ( y ) k L ( H , H ) −→ n → + ∞ . Using (21) together with the fact that Dϕ , P n and L are bounded we get [ P n , Dϕ ( x ) ∗ ] −→ n → + ∞ in operators’ norm, uniformly in x ∈ H on bounded sets . An analogous argument implies that [ P n , Dϕ − ( x ) ∗ ] −→ n → + ∞ in operators’ norm, uniformly in x ∈ H on bounded sets , thus ϕ is an admissible symplectomorphism and we can apply the non-squeezing theorem. Remark . The biggest class of symplectomorphisms (which is not a group)for which our proof of non-squeezing works is the one for which the assump-tions of Proposition 3.2 are fulfilled, namely:
Symp A ( H , ω, h· , ·i ) := ¶ ϕ ∈ Symp ( H , ω ) (cid:12)(cid:12)(cid:12) Dϕ and Dϕ − are bounded; ( I − P n ) ϕ − | P n H −→ n → + ∞ uniformly on bounded sets ;[ P n , Dϕ − ( x ) ∗ ] −→ n → + ∞ in operators’ norm, uniformly in x ∈ H on bounded sets © . Non-squeezing as a critical point theory prob-lem
Let us consider the set of symplectomorphisms
Symp B ( H , ω ) := ¶ ϕ ∈ Symp ( H , ω ) (cid:12)(cid:12)(cid:12) ϕ and ϕ − have bounded differentialsup to the third order. © . Adopting the same notation as in the previous section, our aim is to provethe following.
Proposition 4.1.
Let ( H , ω ) be a Hilbert symplectic space endowed with acompatible inner product, B r the ball centred in with radius r and Z acylinder whose basis lays on a symplectic plane and has symplectic area π .Let ϕ ∈ Symp B ( H ) , if ϕ ( B r ) ⊂ Z and there exists a (PS) c sequence { x n } n ∈ N for A K at level c > , then r ≤ . It is worth to recall that the Hamiltonian K is constructed starting fromthe symplectomorphism ϕ . Remark . Given an arbitrary ϕ ∈ Symp B ( H ) , writing down the actionfunctional A K explicitly is easy, but finding a suitable (PS) c for A K can bean extremely difficult task (if possible). Anyway this task could be madesolvable by requiring additional conditions on ϕ .Starting from now we focus on proving Proposition 4.1. Let us considera map ϕ ∈ Symp B ( H , ω ) and assume that ϕ ( B r ) ⊂ Z , with r > . Underthis assumption we obtain the following lemmata. Lemma 4.1.
Any sequence { x n } n ∈ N ∈ E such that ∇ A K ( x n ) −→ n → + ∞ is H -bounded.Proof. Let us assume that { x n } n ∈ N in unbounded sequence and define thebounded sequence { y n } n ∈ N of elements y n := x n | x n | x + n − x − n − ∇ b K ( x n ) = ∇ A K ( x n ) −→ n → + ∞ we can multiply by | x n | and deduce that y + n − y − n − ∇ b K ( x n ) | x n | −→ n → + ∞ . (22)Let us consider the Hamiltonian defined as I ( z ) := µq ( z ) for any z ∈ H .Using the fact that there exists a constant L such that |∇ K ( z ) − ∇ I ( z ) | ≤ L for any z ∈ H , we get that | ∇ b K ( x n ) | x n | − ∇ I ( y n ) | = | T ∗ Ä ∇ K ( x n ) | x n | − ∇ I ( y n ) ä | ≤ | ∇ K ( x n ) | x n | − ∇ I ( y n ) | L = 1 | x n | |∇ K ( x n ) − ∇ I ( x n ) | L ≤ L | x n | −→ n → + ∞ , hence ∇ b K ( x n ) | x n | − ∇ I ( y n ) −→ n → + ∞ . This in combination with (22) implies that y + n − y − n − ∇ I ( y n ) −→ n → + ∞ . (23)Since I ( z ) = µ k z k with π < µ < π , then (23) is equivalent to the condition Ay n −→ n → + ∞ , with A invertible linear operator (cfr. Example 1.1). Such a condition isfulfilled if and only if | y n | −→ n → + ∞ , but we already know that | y n | = 1 , hencethe initial assumption that { x n } n ∈ N is unbounded leads to a contradiction.44 emma 4.2. There exists a real number η > such that, given any H -bounded (PS) c sequence { x n } n ∈ N of A K for which ∇ L A K ( x n ) is L -infinitesimal,then { x n } n ∈ N admits a (PS) c subsequence { x n k } n ∈ N whose elements are eitherloops taking values in { H < m } + B η ⊂ ϕ ( B r ) or in H \{ H < m } .Proof. According to (12) we can find a value λ > such that { H < m } + B λ ⊂ ϕ ( B r ) . We claim that, up to a subsequence, if x n ( t n ) ∈ H \ ( { H < m } + B λ ) for sometime t n then x n ( t ) / ∈ { H < m } for any t ∈ [0 , (and vice-versa).Indeed, if not there are times ≤ t n < t n ≤ for which x n ( t n ) ∈ H \ ( { H < m } + B λ ) , x n ( t n ) ∈ { H < m } and x n ( t ) ∈ ( { H < m } + B λ ) \{ H < m } for t ∈ [ t n , t n ] . Since x n is absolutely continuous, we get x n ( t n ) = x n ( t n ) + Z t n t n ˙ x n ( s ) ds. The point x n ( t n ) has distance at least λ from x n ( t n ) , therefore λ ≤ | x n ( t n ) − x n ( t n ) | = | Z t n t n ˙ x n ( s ) ds | ≤ Z t n t n | ˙ x n ( s ) | ds. We know that | J ˙ x n + ∇ K ( x n ) | L −→ n → + ∞ (24)and since K is constant on ( { H < m } + B λ ) \{ H < m } , for any t n < s < t n we have that ∇ K ( x n ( s )) = 0 . Thus (24) implies that Z t n t n | ˙ x n ( t ) | dt = Z t n t n | J ˙ x n ( t ) | dt −→ n → + ∞ , but this fact is in contradiction with the estimate λ ≤ Z t n t n | ˙ x n ( s ) | ds ≤ C Z t n t n | ˙ x n ( t ) | dt, with C independent on n , therefore x n cannot travel from H \ ( { H < m } + B λ ) to { H < m } . By setting η := λ we deduce the result.45 emma 4.3. Let { x n } n ∈ N be a H -bounded (PS) c sequence for A K such that ∇ L A K ( x n ) is L -infinitesimal. If all the loops x n have support in the set H \{ H < m } then the level c is not positive.Proof. For any loop x taking values in H \{ H < m } we have that ∇ K ( x ) = 2 ρ ′ ( q ( x ) Qx where Q := dq is the linear map which, on the Hilbert symplectic basis { e i , f i } i ∈ N of H , is written as Q : H → H , ( q , p ) ( q , p ) , ( q i , p i ) N ( q i , p i ) if i ≥ . We are interested in the relation between the Fourier coefficients of a curve x ∈ L ( S , H ) and the ones of the curve Qx .Let x ( t ) = X k ∈ Z e πkJt x k , we write any coefficient x k ∈ H as x k = ( x k , ˆ x k ) , where x k is given by the components of x k in the 2-dimensional plane spannedby { e , f } . We denote with J the two dimensional linear operator mapping e f and f
7→ − e and we define ˆ J := J − J .The image of x via Q is Q ( x ( t )) = Q ( X k e πkJt x k ) = X k Q ( e πkJt x k ) . J and ˆ J commute, thus for any k ∈ Z we deduce that Q ( e πkJt x k ) = Q ( e πkt ( J + ˆ J ) x k ) = Q ( e πktJ e πkt ˆ J x k )= Q ( e πktJ x k + e πkt ˆ J ˆ x k ) = e πktJ x k + 1 N e πkt ˆ J ˆ x k = e πktJ x k + e πkt ˆ J N ˆ x k = e πktJ e πkt ˆ J Q ( x k )= e πkJt Q ( x k ) , hence the k -th Fourier coefficient of Qx is Qx k .Let us consider a H -bounded (PS) c sequence of loops taking values in theset H \{ H < m } ; inferring as in the second part of the proof of Proposition2.2 we obtain that K ( x n ) −→ n → + ∞ c ∈ R ∪ {±∞} uniformly in t. Since the sequence is L -bounded, it has to converge to a constant c ∈ R ,thus we deduce the uniform convergence of ρ ( q ( x n ( t ))) −→ n →∞ c ∈ R and hence of ρ ′ ( q ( x n ( t ))) −→ n →∞ d ∈ R . This implies that ∇ K ( x n ( t )) − dQx n ( t ) −→ n →∞ uniformly in t, and hence ∇ K ( x n ) − dQx n −→ n →∞ in L ( S , H ) . By Lemma 1.1 it follows that |∇ b K ( x n ) − dT ∗ Qx n | = | T ∗ ( ∇ K ( x n ) − dQx n ) | ≤ |∇ K ( x n ) − dQx n | L , thus ∇ b K ( x n ) − dT ∗ Qx n −→ n →∞ in H ( S , H ) .
47n order for x n = ( x − n , x n , x + n ) to define a (PS) c sequence for A K it is necessarythat ∇ a ( x n ) − ∇ b K ( x n ) −→ n →∞ in the H -norm, thus it is necessary that x + n − x − n − dT ∗ Qx n = ∇ a ( x n ) − dT ∗ Qx n −→ n →∞ , in H ( S , H ) . (25)Because of Lemma 1.1 we know that dT ∗ Qx n = 2 dQx n , dT ∗ Qx kn = 2 d π | k | Qx kn for k = 0 . and since ≤ d < π − ǫ for some ǫ > , then ǫ π | x + n − x n | ≤ | ( x + n − x n ) − dT ∗ Q ( x + n − x n ) + | , | x − n | ≤ | x − n + 2 dT ∗ Qx − n | . Therefore (25) is possible only if | x + n − x n | −→ n →∞ and | x − n | −→ n →∞ .We know that ≤ d < π and we shall consider two cases: when d = π andwhen d = π .If d = π then, by Lemma 1.1, in order for { x n } n ∈ N to be a (PS) c sequence itis also necessary that | x n | −→ n →∞ , and since A K ( x n ) = 12 | x + n | − | x − n | − b K ( x n ) , we get that A K ( x n ) −→ n →∞ c, with c ≤ .If d = π , then { x n } n ∈ N approaches curves of the form x ( t ) = e πJt ( q , p , , , . . . ) with k ( q , p , , , . . . ) k = π + ξ , where, by the definition of ρ , we have π + ξ ≤ m . In this case it follows that A K ( x n ) −→ n →∞ π + ξ − m ≤ .
48e are finally ready to prove Proposition 4.1.
Proof.
Let us assume by contradiction that, given r > , the ball B r can besymplectically squeezed into Z by a symplectomorphism ϕ ∈ Symp B ( H ) .By Lemma 4.1 the (PS) c sequence { x n } n ∈ N has to be bounded, moreoverProposition 2.2 and Lemma 4.2 tell us that it is possible to modify the se-quence in such a way that it becomes H -bounded and its elements lay eitherentirely in a fixed domain { H < m } + B η ⊂ ϕ ( B r ) on which A K and A H coincide, or outside of { H < m } .The latter possibility is excluded by Lemma 4.3, hence we can find a H -bounded (PS) c sequence { x n } n ∈ N at positive level for A H . Therefore accord-ing to Proposition 2.1 we can find a (PS) c sequence { ϕ − ( x n ) } n ∈ N for A F atpositive level, but this is in contradiction with Proposition 3.1.49 eferences [Abb01] A. Abbondandolo, Morse Theory for Hamiltonian Systems , Chap-man & Hall/CRC Research Notes in Mathematics , (2001).[AM15] A. Abbondandolo and P. Majer,
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