An example of entropy non-expansive KAM-nondegenerate nearly integrable system
aa r X i v : . [ m a t h . D S ] S e p AN EXAMPLE OF ENTROPY NON-EXPANSIVEKAM-NONDEGENERATE NEARLY INTEGRABLE SYSTEM
DMITRI BURAGO, DONG CHEN AND SERGEI IVANOV
Abstract.
In this paper we present a Hamiltonian perturbation of any com-pletely integrable Hamiltonian system with 2n degrees of freedom ( n ≥ C ∞ small but the resulting flow has positive metric en-tropy and it satisfies KAM non-degeneracy conditions. The key point is thatpositive entropy can be generated in an arbitrarily small tubular neighborhoodof one trajectory. Introduction
We presume that a potential reader interested in this paper is familiar with suchnotions as the symplectic manifolds and Lagrangian subspaces, Hamiltonian vectorfields and flows, Poisson and Lie brackets, first integrals, Lyapunov exponents,metric and topological entropy, integrable systems, and has a basic idea of themain concept of the KAM Theory. We will refresh some of these notion, or to setup notations, and briefly discuss a few more technical aspects of the KAM Theory.Open questions are at the end of the paper.We work on a symplectic manifold (Ω n , ω ) with n ≥
2. A Hamiltonian systemwith 2 n degrees of freedom is called completely integrable if it admits n algebraicallyindependent first integrals which pair-wisely Poisson commute. According to theLiouville-Arnold Theorem (a precise statement can be found in [17]), except for azero measure set, the phase space of a completely integrable system with compactcommon level sets of the integrals is foliated by invariant tori and the motion oneach of these tori is conjugate to some linear flow on a standard torus. Theseinvariant tori are in fact common level sets of the angle variables in the so calledaction-angle coordinates which are constructed in the course of the proof of theLiouville-Arnold Theorem.If one perturbs the Hamiltonian function of a completely integrable system, theresulting Hamiltonian flow is called nearly integrable . For simplicity, we work inthe C ∞ category, though the smoothness can be lower (depending on the dimen-sion). Once the unperturbed system is non-degenerate in a suitable sense, the cel-ebrated Kolmogorov-Arnold-Moser(KAM) theorem [3][16][20] shows that in nearlyintegrable systems, a large amount of invariant tori survive and the dynamics onthese tori is still quasi-periodic. “A large amount” means that the measure of the Mathematics Subject Classification.
Key words and phrases. metric entropy non-expansive maps, KAM theory, Finsler metric, duallens map, Hamiltonian flow, perturbation.The first author was partially supported by NSF grant DMS-1205597. The second author waspartially supported by Dmitri Burago’s Department research fund 42844-1001. The third authorwas partially supported by RFBR grant 17-01-00128. tori which do not survive goes to zero as the size of the perturbation does (the con-crete estimates are of no importance for us here). These invariant tori are called
KAM tori . The tori which survive have “sufficiently irrational” rotation numbers(a certain degree of being
Diophantine , the precise condition is a bit technical andof no importance for this paper).The dynamic outside KAM tori draws a lot of attention. An interesting butrelatively easy (by modern standards, though quite important in its time) questionis whether topological entropy could become positive. This means the presence ofsome hyperbolic dynamics there. Newhouse [21] proved that a C generic Hamil-tonian flow contains a hyperbolic set (a horse-shoe), hence the flow has positivetopological entropy. When we confine our attention to geodesic flows on Riemann-ian manifolds instead of Hamiltonian flows, Kneiper and Weiss [18] proved that a C ∞ generic Riemannian surface admits a hyperbolic set in its geodesic flow. Thisresult was later extended by Contreras [14] to C generic Riemannian manifoldswith dimension at least 2.Then Arnold [4] (in a number of papers followed by papers by Douady [15] andothers) gave examples of what is now known as the Arnold diffusions: there maybetrajectories asymptotic to one invariant torus at one end and then asymptotic toanother torus on the other end. Furthermore, there maybe trajectories which spenda lot of time near one torus, then leave and spend even longer time very close toanother one and so on. This sort of hyperbolicity is, however, very slow. We knowthis by the double-exponential estimates on the transition time due to Nekhoroshev[22].In the presence of canonical invariant measure, topological entropy is not sointeresting: it can (and often does) live on a set of zero measure. To get positivetopological entropy, it suffices to find one Poincar´e-Smale horseshoe (even of zeromeasure). Little is known about is the metric entropy , i.e. the measure theoreticentropy with respect to the Liouville measure on a level set (or to the symplecticvolume on the entire space). Positive metric entropy implies positive topologicalentropy, but not vice versa [8]. Despite the strong interest in nearly integrableHamiltonian systems, what was lacking is understanding whether these systemsadmit positive metric entropy.In this paper we give a positive answer to this question by constructing specificperturbations near any Liouville torus. There are natural questions about genericityof such perturbations or how large the entropy is (See Section 5); these questionsremain open.A dual lens map technology has been recently developed and used in [11] toconstruct a C ∞ Lagrangian perturbation of the geodesic flow on the standard S n ( n ≥
4) such that the resulting flow has positive metric entropy and is entropy non-expansive in the sense of [9]. The ideas grew from the boundary rigidity problems.The tool used in [11] opened a new door towards more interesting results. In somesense, in this paper the dual lens map technique is applied to Lagrangian sub-manifolds in a symplectic manifold rather than to geodesics in a Finsler manifold.By combining the dual lens map techniques with the Maupertuis principle, itis shown in [13] that one can make a C ∞ small Lagrangian perturbation of thegeodesic flow on an Euclidean T n ( n ≥
3) to get positive (though extremely smalldue to [22]) metric entropy. Unlike the case of spheres, the geodesic flows on flattori are KAM-nondegenrate. Therefore in view of KAM theory, the construction in
NTROPY NON-EXPANSIVE KAM SYSTEMS 3 [13] is an improvement of that in [11]. With this result we know that in some regionin the complement of KAM tori, the dynamics of a nearly integrable Hamiltonianflow can be quite stochastic. On the other hand, unlike the construction in [11],the perturbed flows in [13] are entropy expansive.The perturbations in [11] and [13] are constructed for the geodesic flows of thestandard metrics of S n and T n . In this paper we generalize the methods to arbitraryintegrable systems and obtain the following theorem: Theorem 1.1.
Let Φ tH be a completely integrable Hamiltonian flow on a symplecticmanifold Ω = (Ω n , ω ) with n ≥ , and let T ⊂ Ω be a Liouville torus of this flow.Then one can find a C ∞ -small perturbation e H of H such that the resultingHamiltonian flow Φ t e H has positive metric entropy. Furthermore, such perturbationcan be made in an arbitrarily small neighborhood of T and such that the flow isentropy-nonexpansive (see the definition below).Remark . An important class of examples of Hamiltonian flows is providedby (Legendgre transforms of) geodesic flows of Finsler metrics. These are flowson the co-tangent bundle of a base manifold M n with Hamiltonians that are 2-homogeneous and strictly convex on each fiber. If H in Theorem 1.1 belongs tothis class then e H can be chosen from this class too, hence the result can be achievedby perturbing the original Finsler metric on M . We leave to the reader checkingthis fact. For the Riemannian metrics however this remains an open problem.One primary distinction between our examples and those in [13] is the dimen-sion 2. In this case, the 2-dimensional KAM tori separate the 3-dimensional energylevel thus no Arnol’d diffusion is admitted in such systems. Nevertheless we stillget positive metric entropy between these tori.Another improvement is the entropy non-expansiveness. We say a flow Φ t is entropy non-expansive if for any ǫ >
0, there exists an orbit γ such that the set oftrajectories which stay forever within distance no more than ǫ from γ contains anopen invariant set on which the dynamic has positive metric entropy [9]. Basicallyit means that positive metric entropy can be generated in an arbitrarily smallneighborhood of an orbit of the system. The issue attracted a lot of interest, seefor instance [9, 22]. In particular, the first Author introduced this notion in 1988being in mathematical isolation in the former Soviet Union, see [10]. This situationis a bit counter-intuitive since hyperbolic dynamics tends to expand and occupy allspace. In our situation, however, it is generated even near a periodic orbit, meaningthat hyperbolic dynamics is localized in a small neighborhood not only in the phasespace but in the configuration space too. The paper [11] gave a construction of anentropy non-expansive flow however not in the context of the KAM Theory.This paper will be organized in the following ways: in Section 3 we change theproblem of perturbing the Hamiltonian to perturbation of the Poincar´e map. InSection 4 we use a recent result by Berger-Turaev [7] to show how to get positivemetric entropy in a small invariant set by perturbing a family of symplectic maps(including those we get in Section 3) and give a proof of Theorem 1.1. A moredetailed plan of proof can be found in Section 2.3.2. Preliminaries
Hamiltonian flows.
Let (Ω n , ω ) be a 2 n -dimensional symplectic manifoldand H a smooth function on T ∗ M . The Hamiltonian vector field X H is defined as DMITRI BURAGO, DONG CHEN AND SERGEI IVANOV the unique solution to the equation ω ( X H , V ) = dH ( V )for any smooth vector field V on Ω. X H is well-defined due to non-degeneracy of ω .The flow Φ tH on Ω defined by ˙Φ tH = X H is called the Hamiltonian flow on Ω withHamiltonian H . One can easily verify that Φ tH preserves ω and hence the volumeform ω n . Any Hamiltonian flow is locally integrable. To be more specific, we havethe following generalization of Darboux’s theorem [19, Chapter I, Theorem 17.2]: Theorem 2.1 (Carath´eodory-Jacobi-Lie) . Let (Ω n , ω ) be a symplectic manifold.Let a family p , ..., p k of k differentiable functions ( k ≤ n ), which are pairwisePoisson commutative and algebraic independent, be defined in the neighborhood V of a point x ∈ Ω . Then there exists n − p other functions p k +1 , ..., p n , q , ..., q n defined in an open neighborhood U of x in V such that in U we have ω = n X i =1 dq i ∧ dp i . Corollary 2.2.
For any point x ∈ Ω and any Hamiltonian function H , one canfind an open neighborhood U of x and symplectic coordinates ( q , p ) in U such that H | U = p n . Sections and Poincar´e maps.
Throughout the paper Ω = (Ω n , ω ) denotesa symplectic manifold, n ≥ H : Ω → R a smooth Hamiltonian, X = X H theHamiltonian vector field of H , and { Φ tH } t ∈ R the corresponding Hamiltonian flow.A section of the flow { Φ tH } is a (2 n − ⊂ Ωtransverse to the trajectories of the flow. The transversality means that X H isnowhere tangent to Σ. Note that this implies in particular that X H does notvanish on Σ. A section Σ determines the Poincar´e return map which sends a point x ∈ Σ to the first intersection point of the trajectory { Φ tH ( x ) } t> with Σ. This isa partially defined map from Σ to itself.We need to consider a more general situation: given two sections Σ and Σ of { Φ tH } , the associated Poincar´e map is a partially defined map R H : Σ → Σ defined as follows: for x ∈ Σ , R H ( x ) is the first intersection point of the trajec-tory { Φ tH ( x ) } t> with Σ . (If the trajectory does not intersect Σ , then R H ( x ) isundefined). We denote this map by R H, Σ , Σ or by R H when Σ and Σ are clearfrom context.Since Σ and Σ are transverse to the trajectories, R H is defined on an opensubset of Σ and it is a diffeomorphism from this subset to its image in Σ . In thispaper we always choose sections Σ and Σ so that R H, Σ , Σ is a diffeomorphismbetween Σ and Σ . This is achieved by replacing Σ and Σ by suitable smallneighborhoods of some x ∈ Σ and R H ( x ) ∈ Σ .The Hamiltonian induces a number of structures on sections. Here is a list ofstructures and their properties that we need in this paper. For a detailed expositionof relations between flows and their sections, see [23, § Induced measure on sections.
Since the flow Φ tH preserves the canonical sym-plectic volume on Ω, it naturally induces a measure Vol Σ on a section Σ as follows:for a Borel measurable A ⊂ Σ,Vol Σ ( A ) = Vol Ω { Φ tH ( x ) : x ∈ A, t ∈ [0 , } NTROPY NON-EXPANSIVE KAM SYSTEMS 5 where Vol Ω in the right-hand side is the symplectic volume counted with multiplic-ity.One easily sees that Poincar´e maps preserve the induced measure on sections.Furthermore, from Abramov’s formula [1] one sees that the positivity of metricentropy of a Poincar´e return map implies that of the flow: Proposition 2.3.
Let Σ be a section such that the Poincar´e return map R H, Σ , Σ is a diffeomorphism and it has positive metric entropy. Then the flow { Φ tH } haspositive metric entropy. (cid:3) Slicing sections by level sets.
For a section Σ and h ∈ R we denote by Σ h the h -level set of H | Σ : Σ h = { x ∈ Σ : H ( x ) = h } . Since Σ is transverse to the flow, the differential of H | Σ does not vanish. Hence Σ h is a smooth (2 n − ω to Σ h is non-degenerate. Thus Σ h is a symplectic manifold.(2) Let Σ , Σ be two sections such that the Poincar´e map R H : Σ → Σ isa diffeomorphism. Since the flow preserves H , R H sends Σ h to Σ h . Wedenote by R hH the restriction R H | Σ h . One easily sees that R hH preserves thesymplectic form, hence it is a symplectomorphism between Σ h and Σ h .(3) Let Vol Σ h denote the (2 n − h . Astraightforward computation shows that the (2 n − Σ (see above) is determined by the family { Vol Σ h } h ∈ R of symplecticvolumes of level sets:(2.1) Vol Σ ( A ) = Z R Vol Σ h ( A ∩ Σ h ) dh The identity (2.1) and Abramov-Rokhlin entropy formula [2] imply that it sufficesto obtain positive metric entropy for the Poincar´e return map on the slices Σ h .Namely the following holds. Proposition 2.4.
Let Σ be a section such that the Poincar´e return map R H = R H, Σ , Σ is a self-diffeomorphism of Σ . Suppose that there is a set Λ ⊂ R of positiveLebesgue measure such that for every h ∈ Λ , the symplectomorphism R hH : Σ h → Σ h has positive metric entropy. Then R H has positive metric entropy.Remark . The above structures depend on both Σ and H . The measure Vol Σ andthe level sets Σ h are determined by the restriction of H to an arbitrary neighborhoodof Σ. In the proof of Theorem 1.1 we fix suitable sections Σ and Σ and perturb H in a small region between them. The perturbation does not change H near Σ ∪ Σ ,hence the volume forms Vol Σ i on the sections and the splitting of Σ i into symplecticsubmanifolds Σ hi remain the same.However the Poincar´e map can be affected by such perturbations of H . Ourplan is to perturb H in such a way that the resulting Poincar´e return map on asuitable section Σ has an invariant open set where it satisfies the assumptions ofProposition 2.4. DMITRI BURAGO, DONG CHEN AND SERGEI IVANOV
Plan of the proof.
Let Ω, H , T be as in Theorem 1.1. First, by a smallperturbation of H preserving integrability and Liouville tori, we make the flow on T nonvanishing and periodic. Then pick a point y ∈ T and choose a small sectionΣ through y such that y is a fixed point of the Poincar´e return map R H = R H, Σ , Σ .Let Σ be a small neighborhood of y in Σ such that R H | Σ is a diffeomorphismonto its image Σ := R H (Σ ). The remaining perturbations of H occur within atiny neighborhood of a point x lying on the trajectory of y . This guarantees thatthe Poincar´e map R e H = R e H, Σ , Σ is still a diffeomorphism between Σ and Σ .The perturbed Hamiltonian e H should satisfy the following conditions: the Poincar´emap R e H : Σ → Σ has an invariant open set U ∋ y , and the restriction of R e H tothis invariant set has a positive metric entropy. Then, by Proposition 2.3 appliedto U in place of Σ, the flow { Φ t e H } has a positive metric entropy.The construction of e H is divided into two parts. The first part, summarizedin Lemma 4.2, is a construction of a perturbed Poincar´e map e R : Σ → Σ withthe properties desired from the Poincar´e map R e H . The second part, described inSection 3, is a construction of a perturbed Hamiltonian e H which realizes the given e R as its Poincar´e map: e R = R e H .In order to be realizable as a Poincar´e map, the diffeomorphism e R has to satisfythe natural requirements: it should map the slices Σ h to the respective slices Σ h ,and it should preserve the symplectic form on the slices. In fact, in Section 3 weshow that any sufficiently small compactly supported perturbation of R H satisfyingthese requirements is realizable as a Poincar´e map of some perturbed Hamiltonian,see Proposition 3.1.3. Hamiltonian perturbations with prescribed Poincar´e maps
In this section we fulfill the last step of the above plan. We use the notationintroduced in Section 2.2: Ω = (Ω n , ω ) is a symplectic manifold, n ≥ H : Ω → R is a Hamiltonian and { Φ tH } is the corresponding flow, Σ and Σ are sections suchthat the Poincar´e map R H : Σ → Σ is a diffeomorphism. Let y ∈ Σ and let x be a point on the trajectory { Φ tH ( y ) } between Σ and Σ .Let e R be a perturbation of R H with the same properties as R H , namely(1) e R : Σ → Σ is a diffeomorphism;(2) e R preserves H , that is, H ◦ e R = H on Σ . Equivalently, e R (Σ h ) = Σ h forevery h ∈ R ;(3) the restriction of e R to each Σ h preserves the symplectic form.We also assume that e R is C ∞ -close to R H and they coincide outside a small neigh-borhood of our base point y . Our goal is to realize e R as a Poincar´e map of someperturbed Hamiltonian e H . Moreover e H can be chosen C ∞ -close to H and suchthat e H − H is supported is a small neighborhood of x . More precisely, we provethe following. Proposition 3.1.
Let Ω , H , Σ , Σ , y and x be as above. Then for everyneighborhood U of x in Ω there exists a neighborhood V of y in Σ such that, forevery neighborhood H of H in C ∞ (Ω , R ) there exists a neighborhood R of R H in C ∞ (Σ , Σ ) such that the following holds. NTROPY NON-EXPANSIVE KAM SYSTEMS 7
For every e R ∈ R satisfying (1) – (3) above and such that e R = R H outside V ,there exists e H ∈ H such that e H = H outside U , and e R = R e H where R e H : Σ → Σ is the Poincar´e map induced by e H . In the sequel we assume that the neighborhood U in Proposition 3.1 is so smallthat U ∩ (Σ ∪ Σ ) = ∅ where U denotes the closure of U . This guarantees thatthe Hamiltonian remains the same in a neighborhood of Σ ∪ Σ and therefore theinduced structures on Σ and Σ are preserved by the perturbation, see Remark 2.5.In order to prove Proposition 3.1, we first prove the following variant where werealize e R as a Poincar´e map only on one level set H − ( h ). Here we denote by R hH the restriction of R H on Σ h . Proposition 3.2.
Let Ω , H , Σ , Σ , y and x be as above, and let h = H ( x ) .Then for every neighborhood U of x in Ω there exists a neighborhood V h of y in Σ h such that, for every neighborhood H of H in C ∞ (Ω , R ) there exists a neighborhood R h of R hH in C ∞ (Σ h , Σ h ) such that the following holds.For every symplectic e R h ∈ R h such that e R h = R hH outside V h , there exists e H ∈ H such that e H = H on H − ( h ) \ U and e R h = R h e H . Proof of Propositions 3.2.
The proof of Propositions 3.2 is divided into anumber of steps.
Step 1. Localization.
In this step we show that in suffices to prove thepropositions in the canonical case where • Ω = R n = { ( q , p ) : q , p ∈ R n } , with the standard symplectic structure ω = P ni =1 dq i ∧ dp i . • H ( q , p ) = p n . • x is the origin of R n . • Σ = { ( q , p ) : q n = − } and Σ = { ( q , p ) : q n = 1 } .Indeed, our assumptions imply that dH ( x ) = 0. By adding a constant to H we may assume that H ( x ) = 0. By Theorem 2.1 there exist a neighborhood U of x and a symplectic coordinate system ( q , p ), q = ( q , . . . , q n ), p = ( p , . . . , p n ),in U such that p n = H | U and p ( x ) = q ( x ) = 0. In these coordinates we have X H = ∂∂q n .Let ε > Q := { ( q , p ) : | q i | ≤ ε and | p i | ≤ ε for all i } is contained in both U and the union of the trajectories between Σ and Σ . LetΣ − and Σ + be the opposite open faces of Q defined by(3.1) Σ ± = { ( q , p ) : q n = ± ε , | q i | < ε for all i < n, | p i | < ε for all i } . In suffices to prove the propositions for Σ − and Σ + in place of Σ and Σ .Indeed, the assumptions on ε imply that there are diffeomorphic Poincar´e maps R − = R H, Σ ′ , Σ − and R + = R H, Σ + , Σ ′ where Σ ′ and Σ ′ are suitable neighborhoodsof y and R H ( y ) in Σ and Σ , resp. In the statements of the Propositions 3.1and 3.2 we may assume that U ⊂ Q and require that V ⊂ Σ ′ (resp. V h ⊂ Σ ′ ).Then a perturbation e H of H does not change the Poincar´e map outside Σ ′ , andit induces a Poincar´e map e R : Σ ′ → Σ ′ if and only if it induces a a Poincar´e map R − ◦ e R ◦ R − − from Σ − to Σ + . DMITRI BURAGO, DONG CHEN AND SERGEI IVANOV
Thus we may replace Σ and Σ with Σ − and Σ + and assume that U ⊂ Q .Using the coordinates to identify Q with a cube in R n where R n is equipped withthe standard symplectic structure and the standard Cartesian coordinates ( q , p ) =( q , . . . , q n , p , . . . , p n ). The hypersurfaces Σ ± are now subsets of the affine hyper-planes { q n = ± ε } . Applying the symplectic transformation ( q , p ) ( ε − q , p · ε )and multiplying H by a constant we make Σ ± subsets of the hyperplanes { q n = ± } while H is still the coordinate function p n . Now we may extend the structures tothe whole R n and reduce the propositions to the canonical case described above.Throughout the rest of the proof we work (without loss of generality) in thiscanonical setting. Recall that the hypersurfaces Σ i , i = 0 ,
1, are foliated by levelsets Σ hi of H . In our standardized setting we have Σ hi = { ( q , p ) ∈ Σ i : p n = h } , soΣ hi is an (2 n − Step 2.
For each b p = ( b p , . . . , b p n ) ∈ R n , define(3.2) A b p = { ( q , p ) ∈ Σ : p = b p } . Each set A b p is an ( n − h for h = b p n .Moreover A b p is a Lagrangian submanifold of Σ h . We denote by R nh = { ( b p , . . . , b p n ) : b p n = h } . A map e R h satisfying the requirements of Proposition 3.2 maps the partition { A b p } b p ∈ R nh of Σ h to a partition of Σ h into Lagrangian submanifolds e R ( A b p ). The next lemmashows that e R h is uniquely determined by the resulting partition of Σ h . Lemma 3.3.
Let R h , R h : Σ h → Σ h be symplectomorphisms such that R h = R h outside a compact subset of Σ h . Suppose that R h ( A b p ) = R h ( A b p ) for every b p ∈ R nh .Then R h = R h .Proof. Let f = ( R h ) − ◦ R h . The map f is a symplectomorphism from Σ h toitself and it is the identity outside a compact set. Let e , . . . , e n , e n +1 , . . . , e n be the coordinate vectors corresponding to the coordinates q = ( q , . . . , q n ), p =( p , . . . , p n ) in R n . The affine space Σ h is naturally equipped with coordinates( q , . . . , q n − , p , . . . , p n − ). The assumption that R h ( A b p ) = R h ( A b p ) for all b p ∈ R nh implies that f preserves the coordinates p , . . . , p n − . Hence for every ( q , p ) ∈ Σ the partial derivatives of f at ( q , p ) have the form(3.3) ∂f∂p j ( q , p ) = e n + j + v j , j = 1 , . . . , n − , and(3.4) ∂f∂q i ( q , p ) = w i , i = 1 , . . . , n − , where v j , w i belong to the linear span of e , . . . , e n − .Since f preserves the symplectic form ω on every slice { p n = const } of Σ , thevectors v j and w i from (3.3) and (3.4) satisfy ω ( e n + j + v j , w i ) = ω ( e n + j , e i ) = δ ij for all i, j ∈ { , . . . , n − } , where δ ij is the Kronecker delta. Since v j and w i arefrom the linear span of e , . . . , e n − , we have ω ( v j , w i ) = 0. Thus ω ( e n + j , w i ) = δ ij NTROPY NON-EXPANSIVE KAM SYSTEMS 9 for all i, j . Hence w i = e i for all i . Now (3.4) takes the form ∂f∂q i ( q , p ) = e i , i = 1 , . . . , n − . Hence the restriction of f to every subset { q = const } is a parallel translation. Since f is the identity outside a compact set, it follows that f preserves the coordinates q , . . . , q n − and is the identity everywhere. Hence R h = R h . (cid:3) We may assume that the set U where we are allowed to change the Hamiltonianis a cube ( − ε, ε ) n where ε ∈ (0 , V h ⊂ Σ h defined as the projection of U to Σ h , namely(3.5) V h = ( − ε, ε ) n − × {− } × ( − ε, ε ) n − × { h } . We may assume that the neighborhood H of H (see the formulation of theproposition) is so small that every e H ∈ H such that e H = H on H − ( h ) \ U inducesa smooth bijective Poincar´e map R h e H : Σ h → Σ h . Moreover, R h e H = R hH outside V h for every such e H .With Lemma 3.3, Proposition 3.2 boils down to the following statement: Givena sufficiently small perturbation e R h of R hH such that e R h = R hH outside V h , we canconstruct a perturbed Hamiltonian e H ∈ R such that e H = H on H − ( h ) \ U and(3.6) R h e H ( A b p ) = e R h ( A b p ) for all b p ∈ R nh . Step 3.
For each b p ∈ R n , define a Lagrangian affine subspace L b p ⊂ R n by L b p = { ( q , b p ) : q ∈ R n } . The subspaces L b p , where b p ranges over R n , form a foliation of R n . Our plan isto perturb the subfoliation { L b p } b p ∈ R nh and obtain another foliation by Lagrangiansubmanifolds { e L b p } b p ∈ R nh such that(3.7) e L b p ∩ Σ h = A b p and(3.8) e L b p ∩ Σ h = e R h ( A b p )for all b p ∈ R nh , and define the perturbed Hamiltonian e H so that it is constanton each submanifold e L b p . Since the flow (without fixed points) on a level set of aHamiltonian is determined by this set up to a time change, once we fix the level set e H − ( h ), the resulting Poincar´e map on Σ h is independent of other level sets.The next lemma says that this construction solves our problem. We say thata line segment [ x, y ] ⊂ R n is horizontal if it is parallel to coordinate axis of the q n -coordinate. Lemma 3.4.
Let { e L b p } b p ∈ R nh be a foliation by Lagrangian submanifolds satisfying (3.7) and (3.8) . Let e H : R n → R be a smooth function such that (3.9) e H | e L b p = h for all b p ∈ R nh and suppose that e H defines a smooth Poincar´e map R h e H : Σ h → Σ h . Suppose in addition that every horizontal segment intersecting Σ h ∪ Σ h but notintersecting U = ( − ε, ε ) n is contained in one of the submanifolds e L b p . Then R h e H = e R h and e H = H on H − ( h ) \ U .Proof. The key implication of the assumption (3.9) is that every submanifold e L b p is contained in one level set of e H .First we show that e H = H on H − ( h ) \ U . Recall that H = p n and e R (Σ h ) = Σ h .This and (3.7), (3.8), (3.9) imply that e H = H on Σ h ∪ Σ h . Since U is a convexset lying between the hyperplanes Σ and Σ , every point x ∈ H − ( h ) \ U can beconnected to a point y ∈ Σ h ∪ Σ h by a horizontal segment not intersecting U . Bythe assumptions of the lemma the segment [ x, y ] is contained in one level set of e H ,and it is contained in a level set of H since H = p n . Therefore e H ( x ) = H ( x ) forall x ∈ H − ( h ) \ U .Now we show that R h e H = e R h . Fix b p ∈ R nh and consider the leaf e L b p of ourfoliation. By the elementary linear algebra, the property that e L b p is contained ina level set of e H implies that the Hamiltonian vector field X e H is tangent to e L b p .Therefore e L b p is invariant under the flow Φ t e H . By (3.7) and (3.8) it follows that(3.10) R h e H ( A b p ) = L b p ∩ Σ h = e R h ( A b p ) . The fact that e H = H on H − ( h ) \ U implies that R h e H = R hH = e R h outside acompact set. Now with (3.10) at hand we can apply Lemma 3.3 to R h e H and e R h inplace of R h and R h and conclude that R h e H = e R h . (cid:3) It remains to construct a foliation { e L b p } satisfying Lemma 3.4 and such that theresulting Hamiltonian e H is sufficiently close to H in C ∞ . This is achieved in thenext two steps. Step 4.
We begin with a construction of e L b p for a fixed b p ∈ R nh . Throughoutthis step, b p i denotes a fixed real number (the i th coordinate of b p ) and p i , q i arestill coordinate functions, i = 1 , . . . , n .We identify R n with the cotangent bundle T ∗ R n using q i ’s are spatial coordi-nates and p i ’s are coordinates in the fibers of the cotangent bundle. We constructthe desired leaf e L b p as a graph of a closed 1-form e α = e α b p on R n . Recall that agraph of a 1-form is a Lagrangian submanifold of the cotangent bundle if and onlyif the 1-form is closed, see e.g. [12].Define W = ( − ε, ε ) n . The last requirement of Lemma 3.4 prescribes { e L b p } and hence e α outside W . We cover R n by two open half-spaces { q n < ε } and { q n > − ε } and consider e α on these half-spaces with W removed. First consider a1-form α = P b p i dq i (with constant coefficients) and define the restriction of e α to { q n < ε } \ W by(3.11) e α = α on { q n < ε } \ W .
The graph of α is the unperturbed leaf L b p . It satisfies (3.7) and consists of hori-zontal segments, fulfilling the respective part of requirements of Lemma 3.4.Now we define e α on the set { q n > − ε } \ W . In fact, e α on this set is uniquelydetermined by the map e R and the requirement (3.8). Consider the set e A := e R h ( A b p ),the desired intersection of the graph of e α with Σ h . Since e R preserves H and NTROPY NON-EXPANSIVE KAM SYSTEMS 11 the symplectic form in the levels of H , e A is an ( n − h . In the unperturbed case e R h = R hH , e A isan affine subspace Σ h . Hence, if e R h is sufficiently close to R hH in C ∞ , then e A is agraph of a closed 1-form e β defined on the hyperplane { q n = 1 } ⊂ R n . We definethe restriction of e α to { q n > − ε } \ W by(3.12) e α = Π ∗ e β + b p n dq n on { q n > − ε } \ W , where Π is the orthogonal projection from R n to the hyperplane { q n = 1 } . Thegraph of the 1-form defined by (3.12) consists of horizontal segments and satisfies(3.8). Since e R h = R hH outside V (see (3.5)), the definitions (3.11) and (3.12) agreeon the common domain {− ε < q n < ε } \ W .Thus we have defined the desired 1-form e α on R n \ W . Our goal is to extend e α to the whole R n . We need the following lemma. Lemma 3.5. e α defined above is exact on R n \ W .Proof. The statement is trivial if n >
2, since the 1-form is closed and the set R n \ W is simply connected.For n = 2, it suffices to check that the integral of e α over any one cycle goingaround the hole W = ( − ε, ε ) , is zero. We do this for the boundary of the square[ − , . Let s be the side [ − , ×{ } of this square. Since e α = α on the remainingthree sides of the square, it suffices to verify that R s e α = R s α . Each integral is thesigned area between the graph of the respective 1-form and the line { p = 0 } inthe plane Σ h with coordinates ( q , p ). Since one graph is taken to the other bya symplectomophism e R h ◦ ( R hH ) − which is the identity outside the small square( − ε, ε ) , this signed area is preserved. (cid:3) The right-hand sides of (3.11) and (3.12) are closed 1-forms defined on the entire R n . Let f : R n → R and g : R n → R be their antiderivatives. To ensure that f and g depend smoothly on the parameter b p , we choose them so that g (0 , . . . , , −
1) = f (0 , . . . , , −
1) = 0. Observe f = g outside the set ( − ε, ε ) n − × R since (3.11) and(3.12) agree there.We combine f and g using a suitable partition of unity as follows. Fix a smoothfunction µ : R → [0 ,
1] such that µ ( t ) = 1 for all t < − / µ ( t ) = 0 for all t > / e f : R n → R by e f ( q ) = µ ( εq n ) · f ( q ) + (1 − µ ( εq n )) · g ( q ) . Now define e α = d e f everywhere on R n . This definition agrees with (3.11) and (3.12) on their respectivedomains since f and g agree on the set {− ε < q n < ε } \ W .This finishes the construction of the 1-form e α = e α b p for a fixed b p . The graph e L b p of e α b p is a Lagrangian submanifold satisfying the requirements of Lemma 3.4. Step 5.
It remains to show that e H can be chosen to be C ∞ close to H . In orderto prove it we prove the family of Lagrangian submanifolds { e L b p } b p ∈ R nh , constructedin the previous step, is a C ∞ perturbation of the original foliation { L b p } b p ∈ R nh .Going through the constructions of Step 4 one sees that e α b p depends smoothlyon b p . Hence one can define a smooth map F h : H − ( h ) → R n by F h ( q , b p ) = ( q , α b p ( q )) , q ∈ R n , b p ∈ R nh . This map takes each leaf of the original foliation { L b p } to the corresponding La-grangian submanifold e L b p . Note that F h is determined by e R h by means of explicitformulae (involving some inverse functions) and in the unperturbed case (when e R h = R hH ) the resulting map F h is the identity. Therefore F is C ∞ -close to theidentity on any fixed compact set as long as e R is C ∞ -close to R H . We may assumethat the neighborhood R of R hH from which e R h is chosen (see the formulation ofProposition 3.2) is so small that(3.13) k F h − id k C ([ − , n − ×{ h } ) < where the norm of the first derivative is understood as the operator norm.The construction in Step 4 implies that F is the identity on the set { q n < − ε } and F h − id is constant along any horizontal segment not intersecting U . This impliesthat the norm estimate in (3.13) holds in C ( H − ( h )), and this norm estimateimplies that F h is a diffeomorphism from H − ( h ) to its image.Thus e H can be chosen to be C ∞ close to H and we finish the proof of Proposi-tion 3.2.Proposition 3.2 can be applied to prove the following fact which is known infolklore but for which the authors could not find a reference. Proposition 3.6.
Let ϕ : D n → D n , n ≥ , be a symplectomorphism C ∞ -closeto the identity and coinciding with the identity near the boundary. Then there exista smooth family of symplectomorphisms { ϕ t } t ∈ [0 , of D n fixing a neighborhood ofthe boundary and such that ϕ t = ϕ for all t ∈ [0 , ] , ϕ t = id for all t ∈ [ , , andthe family { ϕ t } is C ∞ -close to the trivial family (of identity maps).Proof. Consider Ω = R n +2 = ( q , . . . , q n +1 , p , . . . , p n +1 ) with the standard sym-plectic structure and Hamiltonian H = p n +1 . et Σ and Σ be affine hyperplanesdefined by the equations q n +1 = − q n +1 = 1, resp.We introduce notation ¯p and ¯q for the coordinate n -tuples ( p , . . . , p n ) and( q , . . . , q n ). The Poincar´e map R H : Σ → Σ is given by R H ( ¯q , − , ¯p , p n +1 ) = ( ¯q , , ¯p , p n +1 ) . We define a perturbed map e R : Σ → Σ by e R ( ¯q , − , ¯p ,
0) = ( ϕ ,q ( ¯q , ¯p ) , , ϕ ,p ( ¯q , ¯p ) , ϕ ,q and ϕ ,p are q - and p - coordinates of ϕ . By applying Proposition 3.2to h = 0 we get e H such that e R = R e H . Let H be the Hamiltonian such that H − (0) = e H − (0) (hence e R = R e H = R H ), and H − p n +1 does not depend on p n +1 . Then ∂ H /∂p n +1 = 1 and the return map R H is given by R H ( ¯q , − , ¯p , p n +1 ) = ( ϕ ,q ( ¯q , ¯p ) , , ϕ ,p ( ¯q , ¯p ) , p n +1 ) . Define time-dependent Hamiltonian H t on R n as follows: H t ( ¯q , ¯p ) = H ( ¯q , t, ¯p , . Then the flow on R n generated by H t connects the identity to ϕ . (cid:3) NTROPY NON-EXPANSIVE KAM SYSTEMS 13
Proof of Proposition 3.1.
We deduce Propositions 3.1 from Proposition 3.2by applying it to all h ∈ R and to the corresponding restrictions e R | Σ h in place of e R h . Let e H h denote the resulting perturbed Hamiltonian for a given h . Then thedesired e H in Proposition 3.1 is constructed from the family { e H h } h ∈ R in such a waythat e H − ( h ) = ( e H h ) − ( h ) for every h ∈ R . The resulting Poincar´e map is theprescribed e R since the Hamiltonian flow is determined by level sets up to a timechange.We have only to show that the family of Lagrangian submanifolds e L b p , b p ∈ R n ,constructed in the Step 4 in the proof of Proposition 3.2, forms a foliation of R n .Similar to Step 5, one can define a smooth map F : R n → R n by F ( q , b p ) = ( q , α b p ( q )) , q , b p ∈ R n .F is C ∞ -close to the identity on any fixed compact set as long as e R is C ∞ -close to R H . Furthermore one can slightly adjust the argument in Step 5 to show that F isa diffeomorphism from R n to itself. This finishes the proof of Proposition 3.1.4. Proof of Theorem 1.1
We begin with the following result by Berger-Turaev [7].
Theorem 4.1 ([7]) . For any n ≥ , there is a C ∞ -small perturbation of the identitymap id : D n → D n such that the resulting map is symplecitc and coincides withthe identity map near the boundary and has positive metric entropy.Proof. Theorem A in [7] is proven for n = 1. Though they did not mention inthe theorem whether the perturbed map agrees with the original map near theboundary, Corollary 5 in [5] (see also Corollary 4.8 in [7]) guarantees that they cancoincide near ∂D .To extend the result to n ≥
2, one can do the following. Let ϕ : D → D be theperturbation of the identity constructed for n = 1 and { ϕ t } the family of symplec-tomorphisms constructed in Proposition 3.6 for ϕ = ϕ . Define a diffeomorphismΦ : D × D n − → D × D n − byΦ( x, y ) = ( ϕ | y | ( x ) , y ) , x ∈ D , y ∈ D n − . One easily sees that Φ is a symplectomorphism fixing the neighborhood of theboundary. Since Φ( x, y ) = ϕ ( x ) for all y with | y | ≤ and ϕ has positive metricentropy, so does Φ. (cid:3) Let Ω = (Ω n , ω ) be a symplectic manifold, H : Ω → R a Hamiltonian suchthat the flow { Φ tH } is completely integrable, and T a Liouville torus and x ∈T . By the Liouville-Arnold theorem, there exist action-angle coordinates ( q , p ) =( q , ..., q n , p , .., p n ) near T .These coordinates identify a neighborhood of T with the product T n × D where D ⊂ R n is a small n -dimensional disc and T n = R n / Z n is the standard n -torus.The coordinates p , . . . , p n parametrize D and q , . . . , q n are the standard anglecoordinates on T n (taking values in R / Z ). The Hamiltonian H depends only on p -coordinates, hence it can be regarded as a function on D .The flow Φ tH in these coordinates is governed by the equations(4.1) ( ˙ q i = A i ( p ) := ∂H∂p i ( p )˙ p i = 0 . Thus, along every trajectory the p -coordinates are constant and q -coordinates varylinearly with velocity A i ( p ), i = 1 , . . . , n . We may assume that p = on T and H ( ) = 0.By a small perturbation of the function H = H ( p ) near p = we can satisfythe following conditions: • The flow on T is nonvanishing. (This means that at least one of the numbers ∂H∂p i ( ) is nonzero). • The flow on T is periodic. (It suffices to perturb H so that all numbers ∂H∂p i ( ) are rational). • The system is KAM-nondegenerate at T . In our notation this conditionmeans that the Hessian of H at p = 0 is nondegenegate.We change the coordinates by an action of some matrix from SL ( n, Z ) on T n toassure that the (periodic) flow on T is the flow along the q n -coordinate, that is, A i ( ) = 0 for i = 1 , . . . , n − A n ( ) > H for the modified Hamiltonian and p , q for the modified coordinates. It sufficesto prove the theorem for Hamiltonians and coordinates satisfying the conditionsabove.For h ∈ R , denote D h := { p ∈ D : H ( p ) = h } . Replacing D by a smaller neighborhood of if necessary, we apply the implicitfunction theorem (using the fact that ∂H∂p n ( ) = A n ( ) >
0) and obtain a smoothfamily { f h } h ∈ R of smooth functions f h : R n − → R such that(4.2) p ∈ D h ⇐⇒ p n = − f h ( p , . . . , p n − )for every h ∈ R and p ∈ D . (The minus sign here is introduced to be canceled outlater in (4.4)). We introduce notation ¯p and ¯q for the coordinate ( n − p , . . . , p n − ) and ( q , . . . , q n − ). With this notation (4.2) implies that(4.3) H ( ¯p , − f h ( ¯p )) = h for all ¯p ∈ R n − sufficiently close to the origin and h ∈ R sufficiently close to 0.Differentiating (4.3) with respect to p i we obtain that(4.4) ∂f h ∂p i ( ¯p ) = A i A n ( ¯p , f h ( ¯p )) , i = 1 , ..., n − . Since A i ( ) = 0 for i < n , the origin is a critical point of f h .Now we cut our invariant tubular neighborhood of T by a hypersurfaceΣ = { ( q , p ) : q n = 0 } and consider the resulting Poincar´e return map R = R H : Σ → Σ. The hypersurfaceΣ is naturally identified with T n − × D and parametrized by coordinates ( ¯q , p )where ¯q = ( q , . . . , q n − ) and p = ( p , . . . , p n ). By (4.1), R is given by(4.5) R ( ¯q , p ) = (cid:16) q + A A n ( p ) , . . . , q n − + A n − A n ( p ) , p (cid:17) . Note that the origin of Σ is a fixed point of R .The next lemma is one of the key ingredients of the proof. NTROPY NON-EXPANSIVE KAM SYSTEMS 15
Lemma 4.2.
There exist a diffeomorphism e R : Σ → Σ arbitrarily close to R in C ∞ and such that e R = R outside an arbitrarily small neighborhood of the origin andthe following conditions are satisfied: (1) For every h ∈ R , e R maps the level set Σ h := { x ∈ Σ : H ( x ) = h } to itselfand preserves the symplectic form on this set. (2) There is a small e R -invariant neighborhood of the origin and the restric-tion of e R to this neighborhood has positive metric entropy. Moreover, e R isentropy non-expansive.Remark . In order to speak about metric entropy of e R , we regard Σ with themeasure induced by the symplectic volume on Ω and the original flow Φ tH , see(2.1). One easily sees that any map e R satisfying the first requirement of the lemmapreserves this measure. Proof.
Our nondegeneracy assumption on the Hessian of H ( p ) imply that ¯0 ∈ R n − is a nondegenerate critical point of f . This implies that for all h near0, the function f h has a nondegenerate critical point ¯c ( h ) = ( c ( h ) , . . . , c n − ( h ))depending smoothly on h with ¯c (0) = ¯0 .First we fix h ∈ R sufficiently close to 0 and describe the construction within Σ h .By (4.2), the intersection of Σ h with a suitable neighborhood of the origin isparametrized by a map Γ h : O q × O p → Σgiven by Γ h ( ¯q , ¯p ) = ( ¯q , ¯p , − f h ( ¯p ))where O q and O p are certain neighborhoods of the origin in R n − . Here in the right-hand side we use the coordinates ( ¯q , p ) = ( ¯q , ¯p , p n ) on Σ. Since O p comes from theimplicit function theorem, it can be chosen uniformly in h . Hence we may assumethat ¯c ( h ) ∈ O p . Observe that Γ h preserves the symplectic form, where O q × O p ⊂ R n − is equipped with the standard symplectic form d ¯q ∧ d ¯p = P n − i =1 dq i ∧ dp i .Therefore the restriction of R to the set Γ h ( O q × O p ) ⊂ Σ h is conjugate to asymplectic map G h : O q × O p → R n − ,(4.6) G h = Π ◦ R ◦ Γ h where Π is the standard projection forgetting the last coordinate. For brevity, wedefine m = n − G h so that the resulting map e G h : O q × O p → R m isstill symplectic, it coincides with G h outside a compact subset of O q × O p , has aninvariant neighborhood of this origin and a positive metric entropy in this neigh-borhood.By (4.5) and (4.4), G h can be written in the form(4.7) G h ( ¯q , ¯p ) = ( ¯q + ∇ f h ( ¯p ) , ¯p )where ∇ f h is the Euclidean gradient of f h : R m → R . Notice that G h is the time-1map of the Hamiltonian flow Φ tF h with the Hamiltonian F h given by(4.8) F h ( ¯q , ¯p ) := f h ( ¯p )Our plan is to perturb F h and define e G h as the time-1 map of the flow defined bythe perturbed Hamiltonian. Since ¯c ( h ) is a nondegenerate critical point of f h , by Morse Lemma there exista coordinate chart ¯ P = ( P , . . . , P m ) in O p such that ¯ P vanishes at ¯c ( h ) and(4.9) f h = f h ( ¯c ( h )) + P + · · · + P k − P k +1 − · · · − P m in a neighborhood of ¯c ( h ). We regard P , . . . , P m as functions on O q × Q p by setting P i ( ¯q , ¯p ) = P i ( ¯p ). Then (4.9) is a formula for F h as well, cf. (4.8). Since P , . . . , P m depend only on ¯p -coordinates, they are Poisson commuting. Hence, by Theorem2.1, one can extend this collection of functions to a symplectic coordinate system( ¯ Q , ¯ P ) = ( Q , . . . , Q m , P , . . . , P m ) in a neighborhood of the point ( ¯q , ¯p ) = ( ¯0 , ¯c ( h ))in O q × O p . We may assume that Q , ..., Q m vanish at ( ¯0 , ¯c ( h )).We perturb the Hamiltonian F h in a region U δ := { ¯ P < δ, ¯ Q < δ } where δ is asufficiently small positive number. Let ξ be a smooth function on [0 ,
1] with ξ ≡ , δ/
2] and ξ ≡ δ, ε >
0, define a perturbed Hamiltonian F h,ε,δ by(4.10) F h,ε,δ := F h + ε ξ ( ¯ P ) ξ ( ¯ Q ) ( Q + · · · + Q k − Q k +1 − · · · − Q m ) . Due to the formula (4.9) for F h , the Hamiltonian flow Φ tF h,ε,δ within U δ/ is governedby the following equations in coordinates ( ¯ Q , ¯ P ): ˙ Q i = 2 P i , i ≤ k, ˙ P i = − εQ i , i ≤ k, ˙ Q i = − P i , i > k, ˙ P i = 2 εQ i , i > k. This defines a periodic flow with period 2 π/ √ ε and a fixed point at ¯ P = ¯ Q = ¯ .Outside U δ , the flow Φ tF h,ε,δ coincides with the original flow Φ tF h . We assume that δ is so small that the trajectories of the flow Φ tF h starting in U δ stay within thedomain of ( ¯ Q , ¯ P ) for all t ∈ [0 , tF h,ε,δ .We chose ε so that N := 2 π/ √ ε is an integer. Define(4.11) G h,ε,δ := Φ F h,ε,δ , the time-1 map of the flow determined by the Hamiltonian F h,ǫ,δ . This map isdefined on an open subset of O q × O p containing the closure of U δ (provided that δ is sufficiently small), and it tends to G h in C ∞ as ε → δ ). The disc D δ/ is invariant under G h,ε,δ and the restriction of G h,ε,δ to this disc is N -periodic.Choose a closed disc B ⊂ U δ/ such that the sets B, G h,ε,δ ( B ) , . . . , G N − h,ε,δ ( B ) aredisjoint. This disc is just a sufficiently small ball centered at a non-fixed point of G h,ε,δ . By Theorem 4.1, there exist a symplectomorphism θ : B → B arbitrarily C ∞ -close to the identity fixing a neighborhood of the boundary and having positivemetric entropy. We extend θ to the whole R m by the identity map outside B anduse the same letter θ for its extension to R m .Now define(4.12) e G h = G h,ε,δ ◦ θ. This formula defines e G h in a neighborhood of the closure of U δ . Outside U δ thismap coincides with G h and we extend it by G h to obtain a map e G h : O q × O p → R m . By construction, U δ/ is invariant under e G h , B is invariant under e G Nh , and( e G Nh ) | B = θ | B . Therefore the restriction of e G h to U δ/ has positive metric entropy. NTROPY NON-EXPANSIVE KAM SYSTEMS 17
By choosing ε sufficiently small and θ sufficiently close to the identity, e G h can bemade arbitrarily close to G h in the C ∞ topology.In order to make the perturbed map entropy non-expansive, the construction canbe modified as follows. Instead of working with one disc B , we choose a sequenceof disks { B i } tending to the origin, with diameters going to 0, and such that thesets G kh,ε,δ ( B i ), i ∈ N , k = 0 , . . . , N −
1, are disjoint. We perturb the identity mapwithin each B i as in Theorem 4.1 so that the composition of these perturbations isa C ∞ map θ : R m → R m which is close to the identity. Then the map e G h definedby (4.12) is entropy non-expansive.By the conjugation inverse to (4.6) we transform e G h to a perturbation e R h of R h = R | Σ h . Namely(4.13) e R h = Γ h ◦ e G h ◦ Π | Σ h within the coordinate domain Γ h ( O q × O p ) and e R h coincides with R h outside thisdomain. This finishes the description of the construction within one level set.It remains to show that one can apply the construction simultaneously on all levelsets Σ h , h ∈ R , so that the union of the resulting maps e R h is a diffeomorphism e R : Σ → Σ satisfying the requirements of the lemma.In order to do this, we first construct coordinates ( ¯ Q , ¯ P ) = ( ¯ Q h , ¯ P h ) as abovefor all h from a neighborhood of 0 so that they depend smoothly on h . The ¯ P -coordinates are constructed using the Morse Lemma. In order to make sure thatthey depend smoothly on h , one can apply the Morse-Bott Lemma (a.k.a. theparametric Morse Lemma), see e.g. [6, Theorem 2]. More precisely, to obtain asmooth family { f h } of functions satisfying (4.9), one applies the Morse-Bott Lemmato the function ( ¯p , h ) f h ( ¯p ) − f h ( ¯c ( h )) defined in a neighborhood of 0 in R n − × R .The ¯ Q -coordinates are constructed from ¯ P -coordinates by means of Theorem 2.1.By analyzing the proof of Theorem 2.1 in e.g. [19] one can see that this constructionboils down to explicit formulae involving algebraic computations and solutions ofO.D.E.s, hence it can be made smooth in h in a suitable neighborhood.Having constructed the ( ¯ Q h , ¯ P h )-coordinates for all h ∈ ( − h , h ), we define F h,ε,δ by (4.10) using a small fixed δ and ε = ε ( h ) such that ε ( h ) is a small constantfor | h | < h / ε ( h ) = 0 for | h | > h /
3. Then define e G h by (4.11) and (4.12)using θ = θ h depending on h as follows: { θ h } is a smooth family C ∞ -close toa constant one, θ h is a fixed map θ as above for | h | < h /
3, and θ h = id for | h | > h /
3. The existence of such a family is guaranteed by Proposition 3.6.Finally, define e R h by (4.13). The union of maps e R h forms a self-diffeomorphismof the set { x ∈ Σ : H ( x ) ∈ ( − h , h ) } . By construction, this diffeomorphismcoincides with R on the set of x such that | H ( x ) | ∈ (2 h / , h ). We extend it to adiffeomorphism e R : Σ → Σ by setting e R = R on the remaining part of Σ.The resulting map e R has an invariant neighborhood { P < δ/ , Q < δ/ , | H | Here we briefly discuss a few open problems, some of them are mentioned above.1. In case of the geodesic flow on a Riemannian manifold, we do not know how tomake the perturbation Riemannian. This seems to be quite an intriguing problem.2. How large entropy can be generated depending on the size of perturbation (anyestimates would certainly involve some characteristics of the unperturbed system)?Probably some (very non-sharp) lower bounds can be obtain by a careful analysis ofthe proof. As for the upper bounds, we suspect they should be double-exponentiallike Nekhoroshev estimates.3. Our construction is very specific and non-generic. What abut a generic per-turbation? 6. Acknowledgments We are grateful to Vadim Kaloshin, Anatole Katok, Federico Rodriguez Hertz,Leonid Polterovich and Yakov Sinai for useful discussions. References [1] L. Abramov, On the entropy of a flow (Russian), Dokl. Akad. Nauk SSSR (1959) 873–875.[2] L. M. Abramov and V. A. Rokhlin, The entropy of a skew product of measure- preservingtransformations , Amer. Math. Soc. Transl. Ser. 2, (1966), 255–265.[3] V. Arnold, Proof of a Theorem by A. N. Kolmogorov on the invariance of quasi-periodicmotions under small perturbations of the Hamiltonian, Russian Math. Survey 18 (1963),13–40.[4] V. Arnold, Instabilities in dynamical systems with several degrees of freedom , Sov. Math.Dokl. 5 (1964), 581-585.[5] A. Avila. On the regularization of conservative maps, Acta Math., 205(1):5–18, 2010.[6] A. Banyaga, D. E. Hurtubise, A proof of the Morse-Bott Lemma , Expo. Math. (2004),no. 4, 365–373.[7] P. Berger, D. Turaev, On Hermans Positive Entropy Conjecture , Adv. Math. 349 (2019),1234-1288.[8] A. Bolsinov, I. Taimanov, Integrable geodesic flows on the suspensions of toric automor-phisms, Proceedings of the Steklov Institute of Math. (2000), 42–58.[9] R. Bowen, Extropy-expansive maps , Trans. Amer. Math. Soc. 164 (1972), 323-331.[10] D. Burago, A new approach to the computation of the entropy of geodesic flow and similardynamical systems , Soviet Math. Doklady 37 (1988), 1041–1044.[11] D. Burago, S. Ivanov. Boundary distance, lens maps and entropy of geodesic flows of Finslermetrics , Journal of Geometry and Topology , (2016), 469–490.[12] A. Cannas da Silva, Lectures on symplectic geometry , Lecture Notes in Mathematics, 1764.Springer-Verlag, Berlin, 2001. xii+217 pp.[13] D. Chen, Positive metric entropy arises in some nondegenerate nearly integrable systems ,Journal of Modern Dynamics, Volume 11 (2017), 43-56.[14] C. Contreras, Geodesic flows with positive topological entropy, twist map and hyperbolicity ,Annals of Mathematics, Second Series, Vol. 172, No. 2 (2010), 761-808[15] R. Douady, Stabilit´e ou instabilit´e des point fixes elliptiques, Ann. Sci. ´Ecole Norm. Sup. (4),21 (1988), 1-46.[16] A. Kolmogorov, On the conservation of conditionally periodic motions under small pertur-bation of the Hamiltonian , Dokl. Akad. Nauk SSSR 98 (1954), 525-530.[17] A. Katok, B. Hasselblat, Introduction to the Modern Theory of Dynamical Systems , Vol. 54.Cambridge university press (1997). NTROPY NON-EXPANSIVE KAM SYSTEMS 19 [18] G. Knieper, H. Weiss, A surface with positive curvature and positive topological entropy , J.Differential Geom. 39 (1994), no. 2, 229-249.[19] P. Libermann, C.-M. Marle, Symplectic Geometry and Analytical Mechanics , Vol. 35,Springer Science and Business Media (2012).[20] J. Moser, On invariant curves of area-preserving mappings of an annulus , Nach. Akad. Wiss.G¨ottingen, Math. Phys. Kl. II 1 (1962), 1-20.[21] S. Newhouse, Quasi-elliptic periodic points in conservative dynamical systems , AmericanJournal of Mathematics 99.5 (1977): 1061-1087.[22] S. Newhouse, Continuity Properties of Entropy , Annals of Mathematics, Second Series, Vol.129, No. 1 (1989), 215-235.[23] K. Petersen, Ergodic theory . Cambridge University Press, Cambridge, 1983. Dmitri Burago: Pennsylvania State University, Department of Mathematics, Univer-sity Park, PA 16802, USA E-mail address : [email protected] Dong Chen: Pennsylvania State University, Department of Mathematics, UniversityPark, PA 16802, USA E-mail address : [email protected] Sergei Ivanov, St.Petersburg Department of Steklov Mathematical Institute, Rus-sian Academy of Sciences, Fontanka 27, St.Petersburg 191023, Russia E-mail address ::