Announcement of "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 ANNOUNCEMENT OF “AN EXAMPLE OF ENTROPYNON-EXPANSIVE KAM-NONDEGENERATE NEARLY INTEGRABLESYSTEM”
DMITRI BURAGO, DONG CHEN AND SERGEI IVANOV
Abstract.
This paper is an announcement of a result followed with explanations of someideas behind. The proofs will appear elsewhere. Our goal is to construct a Hamiltonianperturbation of any completely integrable Hamiltonian system with 2n degrees of freedom( n ≥ C ∞ small but the resulting flow has positive metric entropyand it satisfies KAM non-degeneracy conditions. The key point is that positive entropy canbe generated in an arbitrarily small tubular neighborhood of one trajectory. Introduction
This paper is just an announcement of a result followed by explanations of some ideasbehind it and brief discussions. The detailed proofs will appear elsewhere.We presume that a potential reader interested in this paper is familiar with such notionsas the symplectic manifolds and Lagrangian subspaces, Hamiltonian vector fields and flows,Poisson and Lie brackets, first integrals, Lyapunov exponents, metric and topological entropy,integrable systems, and has a basic idea of the main concept of the KAM Theory. We willrefresh some of these notion, to set up notations, and briefly discuss a few more technicalaspects of the KAM Theory. Open questions are also at the end of the paper.We work on a symplectic manifold (Ω n , ω ) with n ≥
2. A Hamiltonian system with 2 n degrees of freedom is called completely integrable if it admits n algebraically independent firstintegrals which pair-wisely Poisson commute. According to the Liouville-Arnold Theorem(a precise statement can be found in [15]), except for a zero measure set, the phase space ofa completely integrable system with compact common level sets of the integrals is foliatedby invariant tori and the motion on each of these tori is conjugate to some linear flow on astandard torus. These invariant tori are in fact common level sets of the angle variables inthe so called action-angle coordinates which are constructed in the course of the proof of theLiouville-Arnold Theorem. Mathematics Subject Classification.
Key words and phrases. metric entropy non-expansive maps, KAM theory, Finsler metric, duel lens map,Hamiltonian flow, perturbation.The first author was partially supported by NSF grant DMS-1205597. The second author was partiallysupported by Dmitri Burago’s Department research fund 42844-1001. The third author was partially sup-ported by RFBR grant 17-01-00128. f one perturbs the Hamiltonian function of a completely integrable system, the result-ing Hamiltonian flow is called nearly integrable . For simplicity, we work in the C ∞ cate-gory, though the smoothness can be lower (depending on the dimension). Once the un-perturbed system is non-degenerate in a suitable sense, the celebrated Kolmogorov-Arnold-Moser(KAM) Theorem [3][14][18] shows that in nearly integrable systems, a large amountof invariant tori survive and the dynamics on these tori is still quasi-periodic. “A largeamount” means that the measure of the tori which do not survive goes to zero as the sizeof the perturbation does (the concrete estimates are of no importance for us here). Theseinvariant tori are called KAM tori . The tori which survive have “sufficiently irrational” rota-tion numbers (a certain degree of being
Diophantine , the precise condition is a bit technicaland of no importance for this paper).The dynamic outside KAM tori draws a lot of attention. An interesting but relatively easy(by modern standards, though quite important at its time) question is whether topologicalentropy could become positive. This means the presence of some hyperbolic dynamics there.Newhouse [19] proved that a C generic Hamiltonian flow contains a hyperbolic set (a horse-shoe), hence the flow has positive topological entropy. When we confine our attention togeodesic flows on Riemannian manifolds instead of Hamiltonian flows, Kneiper and Weiss[16] proved that a C ∞ generic Riemannian surface admits a hyperbolic set in its geodesicflow. This result was later extended by Contreras [12] to C generic Riemannian manifoldswith dimension at least 2.Then Arnold [4] (in a number of papers followed by ones by Douady [13] and others)gave examples of what is now known as the Arnold diffusions: There maybe trajectoriesasymptotic to one invariant torus at one end and then asymptotic to another torus on theother end. Furthermore, there maybe trajectories which spend a lot of time near one torus,then leave and spend even longer time very close to another one and so on. This sort ofhyperbolicity is, however, very slow. We know this by the double-exponential estimates onthe transition time due to Nekhoroshev [20].In the presence of the canonical invariant measure, topological entropy is not so interesting:it can (and often does) live on a set of zero measure. To get positive topological entropy, itsuffices to find one Poincar´e-Smale horseshoe (even of zero measure). Little is known about isthe metric entropy , i.e. the measure theoretic entropy with respect to the Liouville measureon a level set (or to the symplectic volume on the entire space). Positive metric entropyimplies positive topological entropy, but not vice versa [7]. Despite of the strong interestin nearly integrable Hamiltonian systems, what was lacking is understanding whether thesesystems admit positive metric entropy.In this paper we give a positive answer to this question by constructing a specific per-turbations near any Liouville torus. There are natural questions about genericity of suchperturbations or how large the entropy is (See Section 5); these questions remain open.A dual lens map technology has been recently developed and used in [10] to constructa C ∞ Lagrangian perturbation of the geodesic flow on the standard S n ( n ≥
4) such thatthe resulting flow has positive metric entropy and is entropy non-expansive in the sense of[8]. The ideas grew from the boundary rigidity problems. The tool used in [10] opened anew door towards more interesting results. In some sense, in this paper the dual lens maptechnique is applied to Lagrangian sub-manifolds in a symplectic manifold rather than togeodesics in a Finsler manifold. y combining the dual lens map techniques with the Maupertuis principle, it is shownin [11] that one can make a C ∞ small Lagrangian perturbation of the geodesic flow on anEuclidean T n ( n ≥
3) to get positive (though extremely small due to [20]) metric entropy.Unlike the case of spheres, the geodesic flows on flat tori are KAM-nondegenrate. Thereforein view of KAM theory, the construction in [11] is an improvement of that in [10]. Withthis result we know that in some region in the complement of KAM tori, the dynamics of anearly integrable Hamiltonian flow can be quite stochastic. On the other hand, unlike theconstruction in [10], the perturbed flows in [11] are entropy expansive.The perturbations in [10] and [11] are constructed for the geodesic flows of the standardmetrics of S n and T n . In this paper we generalize the methods to arbitrary integrable systemsand obtain the following theorem: Theorem 1.1.
Let Φ tH be a completely integrable Hamiltonian flow on a symplectic manifold 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 resulting Hamiltonianflow Φ t e H has positive metric entropy. Furthermore, such perturbation can be made in anarbitrarily small neighborhood of T and such that the flow is entropy-nonexpansive (see thedefinition below).Remark . An important class of examples of Hamiltonian flows is provided by (Legendgretransforms of) geodesic flows of Finsler metrics. These are flows on the co-tangent bundle ofa base manifold M n with Hamiltonians that are 2-homogeneous and strictly convex on eachfiber. If H in Theorem 1.1 belongs to this class then e H can be chosen from this class too,hence the result can be achieved by perturbing the original Finsler metric on M . We leave tothe reader checking this. For the Riemannian metrics however this remains an open problem.One primary distinction between our examples and those in [11] is the dimension 2. In thiscase, the 2-dimensional KAM tori separate the 3-dimensional energy level thus no Arnol’ddiffusion is admitted in such systems. Nevertheless we still get positive metric entropybetween 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 of trajectories whichstay forever within distance no more than ǫ from γ contains an open invariant set on whichthe dynamic has positive metric entropy [8]. Basically it means that positive metric entropycan be generated in an arbitrarily small neighborhood of an orbit of the system. The issueattracted a lot of interest, see for instance [8][20]. In particular, the first Author introducedthis notion in 1988 being in mathematical isolation in the former Soviet Union, see [9]. Thissituation is a bit counter-intuitive since hyperbolic dynamics tends to expand and occupyall space. In our situation, however, it is generated even near a periodic orbit, meaning thathyperbolic dynamics is localized in a small neighborhood not only in the phase space but inthe configuration space too. The paper [10] gave a construction of an entropy non-expansiveflow however not in the context of the KAM Theory.2. Preliminaries
Hamiltonian flows.
Throughout the paper Ω = (Ω n , ω ) denotes a symplectic man-ifold, n ≥ H : Ω → R a smooth Hamiltonian, X = X H the Hamiltonian vector field f H , and { Φ tH } t ∈ R the corresponding Hamiltonian flow. Any Hamiltonian flow is locallyintegrable. To be more specific, we have the following generalization of Darboux’s theorem[17, Chapter I, Theorem 17.2]: Theorem 2.1 (Carath´eodory-Jacobi-Lie) . Let (Ω n , ω ) be a symplectic manifold. Let afamily p , ..., p k of k differentiable functions ( k ≤ n ), which are pairwise Poisson commutativeand algebraic independent, be defined in the neighborhood V of a point x ∈ Ω . Then thereexists 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 can find an openneighborhood U of x and symplectic coordinates ( q , p ) in U such that H | U = p n . Sections and Poincar´e maps.
Given two sections Σ and Σ of { Φ tH } , we can definethe associated Poincar´e map map R H : Σ → Σ by taking intersection of Σ and orbits fromΣ . In this paper 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 small neighborhoodsof some x ∈ Σ and R H ( x ) ∈ Σ .Since the flow Φ tH preserves the canonical symplectic volume on Ω, it naturally induces ameasure 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 , } where Vol Ω in the right-hand side is the symplectic volume counted with multiplicity.One easily sees that Poincar´e maps preserve the induced measure on sections. Further-more, from Abramov’s formula [1] one sees that the positivity of metric entropy of a Poincar´ereturn 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 dif-feomorphism and it has positive metric entropy. Then the flow { Φ tH } has positive metricentropy. (cid:3) 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 } . Σ h is a smooth (2 n − ω . Moreover, for any diffeomorphic Poincar´e map R H : Σ → Σ , R H sendsΣ h to Σ h . We denote by R hH the restriction R H | Σ h and it is a symplectomorphism.The Abramov-Rokhlin entropy Formula [2] implies that in order to prove Proposition 2.3,it suffices to 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 aself-diffeomorphism of Σ . Suppose that there is a set Λ ⊂ R of positive Lebesgue measuresuch that for every h ∈ Λ , the symplectomorphism R hH : Σ h → Σ h has positive metric entropy.Then R H has positive metric entropy. .3. Plan of the proof.
The construction of e H is divided into two parts. The first part,summarized in Lemma 4.1, 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 in Section 3,is a construction of a perturbed Hamiltonian e H which realizes the given e R as its Poincar´emap: e R = R e H .3. Hamiltonian perturbations with prescribed Poincar´e maps
In this section we show that certain perturbations of Poincar´e map can be realized as thePoincar´e map of a perturbed Hamiltonian flow. We use the notation introduced in Section2.2: Ω = (Ω n , ω ) is a symplectic manifold, n ≥ H : Ω → R is a Hamiltonian and { Φ tH } isthe corresponding flow, Σ and Σ are sections such that the Poincar´e map R H : Σ → Σ isa 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 for every 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 neighborhood ofour base point y . Our goal is to realize e R as a Poincar´e map of some perturbed Hamiltonian e H . Moreover e H can be chosen C ∞ -close to H and such that e H − H is supported is a smallneighborhood of x . More precisely, we prove the following. Proposition 3.1.
Let Ω , H , Σ , Σ , y and x be as above. Then for every neighborhood U of x in Ω there exists a neighborhood V of y in Σ such that, for every neighborhood H of H in C ∞ (Ω , R ) there exists a neighborhood R of R H in C ∞ (Σ , Σ ) such that the followingholds.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 mapinduced by e H . In order to prove Proposition 3.1, we firstly prove the following variant of Proposition 3.1where we realize e R as a Poincar´e map only on one level set H − ( h ). Proposition 3.2.
Let Ω , H , Σ , Σ , y and x be as above, and let h = H ( x ) . Then forevery neighborhood U of x in Ω there exists a neighborhood V h of y in Σ h such that, forevery 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 suchthat 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 a numberof steps.
Step 1.
By Theorem 2.1, it suffices to prove the propositions in the canonical case whereΩ = R n = { ( q , p ) : q , p ∈ R n } , ω = d q ∧ d p , H ( q , p ) = p n , x = ( , ), Σ = { ( q , p ) : q n = } and Σ = { ( q , p ) : q n = 1 } . Throughout the rest of the proof we work in this canonicalsetting. Step 2.
For each b p = ( b p , . . . , b p n ) ∈ R n , define A b p := { ( q , p ) ∈ Σ : p = b p } . Eachset 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 apartition of Σ h into Lagrangian submanifolds e R ( A b p ). The next lemma shows that e R h isuniquely 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 acompact 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 . 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 is toperturb the subfoliation { L b p } b p ∈ R nh and obtain another foliation by Lagrangian submanifolds { e L b p } b p ∈ R nh such that(3.1) e L b p ∩ Σ h = A b p and(3.2) 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 constant on eachsubmanifold e L b p . The next lemma says that this construction solves our problem. We saythat a 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 (3.1) and (3.2) . Let e H : R n → R be a smooth function such that (3.3) 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 not intersecting 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 . It remains to construct a foliation { e L b p } satisfying Lemma 3.4 and such that the resultingHamiltonian e H is sufficiently close to H in C ∞ . This is achieved in the next two steps. Step 4.
Construction of { e L b p } . We firstly identify R n with the cotangent bundle T ∗ R n using q i ’s are spatial coordinates and p i ’s are coordinates in the fibers of the cotangentbundle. Then we construct the desired leaves e L b p as graphs of some closed 1-forms e α = e α b p on R n . Step 5.
Show that e H can be chosen to be C ∞ close to H and we finish the proof ofProposition 3.2. .2. Proof and application of Proposition 3.1.
Proposition 3.1 can be proved by ap-plying Proposition 3.2 to all h ∈ R and to the corresponding restrictions e R | Σ h in place of e R h . Let { e H h } h ∈ R be the family of the Hamiltonians generated in Proposition 3.2, then thedesired e H in Proposition 3.1 is constructed via e H − ( h ) = ( e H h ) − ( h ) for every h ∈ R .We apply Proposition 3.1 to prove the following fact which is known in folklore but forwhich the authors could not find a reference. Proposition 3.5.
Let ϕ : D n → D n , n ≥ , be a symplectomorphism C ∞ -close to theidentity and coinciding with the identity near the boundary. Then there exist a smoothfamily of symplectomorphisms { ϕ t } t ∈ [0 , of D n fixing a neighborhood of the boundary andsuch that ϕ t = ϕ for all t ∈ [0 , ] , ϕ t = id for all t ∈ [ , , and the family { ϕ t } is C ∞ -closeto the trivial family (of identity maps). Ideas of the Proof of Theorem 1.1
Let Ω, H , T be as in Theorem 1.1. By the Liouville-Arnold theorem, there exist action-angle coordinates ( q , p ) = ( q , ..., q n , p , .., p n ) near T . We may assume that p = on T , H ( ) = 0 and X H is colinear to ∂/∂p n on T . By perturbing the function H = H ( p ) near p = if necessary, we also assume the system is KAM-nondegenerate at T , and the flow on T is nonvanishing and periodic.Pick a point y ∈ T and choose a small section Σ through y . Let Σ be a small neigh-borhood of y in Σ such that the Poincar´e return map R = R H, Σ , Σ restricted on Σ is adiffeomorphism onto its image Σ := R (Σ ). The sections Σ , Σ are naturally identifiedwith open sets in T n − × D and parametrized by coordinates ( ¯q , p ) where ¯q = ( q , . . . , q n − )and p = ( p , . . . , p n ). R is given by(4.1) R ( ¯q , p ) = (cid:16) q + ∂H/∂p ∂H/∂p n ( p ) , . . . , q n − + ∂H/∂p n − ∂H/∂p n ( p ) , p (cid:17) . Note that the origin of Σ is a fixed point of R . Then we get Theorem 1.1 from Proposition3.1 and the following lemma: Lemma 4.1.
There exists a diffeomorphism e R : Σ → Σ arbitrarily close to R in C ∞ andsuch that e R = R outside an arbitrarily small neighborhood of the origin and the followingconditions are satisfied: (1) For every h ∈ R , e R maps the level set Σ h := { x ∈ Σ : H ( x ) = h } to itself andpreserves the symplectic form on this set. (2) There is a small e R -invariant neighborhood of the origin and the restriction of e R tothis neighborhood has positive metric entropy. Moreover, e R is entropy non-expansive. The perturbation of H occur within a tiny neighborhood of a point x lying on the trajec-tory of y . This guarantees that the Poincar´e map R e H = R e H, Σ , Σ is still a diffeomorphismbetween Σ and Σ .In the proof of Lemma 4.1 we use Proposition 3.5, Morse-Bott Lemma ([5]) and thefollowing result from Berger-Turaev: Theorem 4.2 (Berger-Turaev [6]) . For any n ≥ , there is a C ∞ -small perturbation of theidentity map id : D n → D n such that the resulting map is symplecitc and coincides with theidentity map near the boundary and has positive metric entropy. . Some open problems
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 to make theperturbation Riemannian. This seems to be quite an intriguing problem.2. How large entropy can be generated depending on the size of perturbation (any esti-mates would certainly involve some characteristics of the unperturbed system)? Probablysome (very non-sharp) lower bounds can be obtain by a careful analysis of the proof. Asfor the upper bounds, we suspect they should be double-exponential alike Nekhoroshev es-timates.3. Our construction is very specific and non-generic. What about a generic perturbation?
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Dmitri Burago: Pennsylvania State University, Department of Mathematics, UniversityPark, 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, RussianAcademy of Sciences, Fontanka 27, St.Petersburg 191023, Russia
E-mail address : [email protected]@pdmi.ras.ru