The Calabi invariant for Hamiltonian diffeomorphisms of the unit disk
aa r X i v : . [ m a t h . S G ] F e b The Calabi invariant for Hamiltonian diffeomorphisms ofthe unit disk
Benoît JolyFebruary 2021
Abstract
In this article, we study the Calabi invariant on the unit disk usually defined oncompactly supported Hamiltonian diffeomorphisms of the open disk. In particular weextend the Calabi invariant to the group of C diffeomorphisms of the closed disk whichpreserves the standard symplectic form. We also compute the Calabi invariant for somediffeomorphisms of the disk which satisfies some rigidity hypothesis. Contents Ą Cal and Ą Cal . . . . . . . . . . . . . . . . . . . . . . . . 154.2 Continuity of Ą Cal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Computation of
Cal in some rigidity cases 225.1 A simple case of C rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2 C -rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Examples 266.1 An example of C rigidity, the super Liouville type . . . . . . . . . . . . . 276.2 An example of C -rigidity, the non Bruno type . . . . . . . . . . . . . . . 28References 28 Let us begin with some basic definitions of symplectic geometry.Let us consider p M n , ω q a symplectic manifold , meaning that M is an even dimen-sional manifold equipped with a closed non-degenerate differential 2-form ω called the symplectic form . We suppose that π p M q “ and that ω is exact, meaning that there xists a -form λ , called a Liouville form , which satisfies dλ “ ω .Let us consider a time-dependent vector field p X t q t P R defined by the equation dH t “ ω p X t , . q , (1)where H : R ˆ M Ñ R p t, x q ÞÑ H t p x q is a smooth function -periodic on t , meaning that H t ` “ H t for every t P R . The function H is called a Hamiltonian function . If the vector field p X t q t P R is complete, it induces afamily p f t q t P R of diffeomorphisms of M that preserve s ω , also called symplectomorphisms or symplectic diffeomorphisms , satisfying the equation BB t f t p z q “ X t p f t p z qq . In particular the family I “ p f t q t Pr , s defines an isotopy from id to f . The map f iscalled a Hamiltonian diffeomorphism . It is well known that the set of Hamiltonian diffeo-morphisms of a symplectic manifold M is a group which we denote Ham p M, ω q , we referto [28] for more details.Let us consider p M, ω q a symplectic manifold which is boundaryless, π p M q “ andsuch that ω is exact. We say that H is a compactly supported Hamiltonian function ifthere exists a compact set K Ă M such that H t vanishes outside K for every t P R . Acompactly supported Hamiltonian function induces a compactly supported Hamiltoniandiffeomorphism f . Such a map is equal to the identity outside a compact subset of M .Let us consider a compactly supported Hamiltonian diffeomorphism f and λ a Liouvilleform on M . The form f ˚ λ ´ λ is closed because f is symplectic but we have more, it isexact. More precisely there exists a unique compactly supported function A f : M Ñ R ,also called action function , such that dA f “ f ˚ λ ´ λ. In the literature the
Calabi invariant
Cal p f q of f is defined as the mean of the function A f and we have Cal p f q “ ż M A f ω n , (2)where ω n “ ω ^ ... ^ ω is the volume form induced by ω , see [28] for more details. We willprove later that the number Cal p f q does not depend on the choice of λ .Let us give another equivalent definition of the Calabi invariant for a compactly sup-ported Hamiltonian diffeomorphism f . We note H a compactly supported Hamiltonianfunction defining f . The Calabi invariant of f can also be defined by the equation Cal p f q “ p n ` q ż ż M H t ω n dt. (3)To prove that ş M A f ω n does not depend on the choice of the Liouville form λ , one mayuse the fact that the action function A f satisfies A f p z q “ ż p ι p X s q λ ` H s q ˝ f s p z q ds, (4) here p X s q s P R is the time dependent vector field induced by H by equation (1) and p f s q s P R is the isotopy induced by the vector field p X s q s P R . Moreover, ş ş M H t ω n dt does not de-pend on the compactly supported Hamiltonian function H defining f .The function Cal defines a real valued morphism on the group of compactly supportedHamiltonian diffeomorphisms of M and thus it is a conjugacy invariant. It is an importanttool in the study of difficult problems such as the description of the algebraic structure ofthe groups Ham p M, ω q : A.Banyaga proved in [3] that the kernel of the Calabi invariantis always simple, which means that it does not contain nontrivial normal subgroups.In this article, we study the case of the dimension two and more precisely the caseof the closed unit disk which is a surface with boundary. We denote by || . || the usualEuclidian norm on R , by D the closed unit disk and by S its boundary. The group of C orientation preserving diffeomorphisms of D will be denoted by Diff ` p D q . We consider Diff ω p D q the group of C symplectomorphisms of D which preserve the normalized stan-dard symplectic form ω “ π du ^ dv, written in cartesian coordinates p u, v q . In the caseof the disk, the group Diff ω p D q is contractile, see [20] for a proof, and coincides with thegroup of Hamiltonian diffeomorphisms of D . Moreover, the -form ω induces the Lebesgueprobability measure denoted by Leb and the symplectic diffeomorphisms are the C dif-feomorphisms of D which preserve the Lebesgue measure and the orientation.Let us begin by the case of the unit open disk ˚ D . The open disk is boundaryless hencewe already have two equivalent definitions of the Calabi invariant given by equations 2and 3 on the set of compactly supported symplectic diffeomorphisms of ˚ D . Let us give athird one. A. Fathi in his thesis [12] gave a dynamical definition which is also describedby J.-M. Gambaudo and É. Ghys in [16]: if we consider an isotopy I “ p f t q t Pr , s from id to f , there exists an angle function Ang I : ˚ D ˆ ˚ D z ∆ Ñ R where ∆ is the diagonal of ˚ D ˆ ˚ D such that for each p x, y q P ˚ D ˆ ˚ D z ∆ , the quantity π Ang I p x, y q is the variation ofangle of the vector f t p y q ´ f t p x q between t “ and t “ . If f is a compactly supported C symplectic diffeomorphism then this angle function is integrable (see section 3) and itholds that Cal p f q “ ż ˚ D ˆ ˚ D z ∆ Ang I p x, y q d Leb p x q d Leb p y q , (5)where the integral does not depend on the choice of the isotopy.In this article we will give an answer to the following question. Question 1.
How to define an extension of the Calabi invariant to the group
Diff ω p D q ? M. Hutchings [23] extended the definition given by equation 3 to the C symplecticdiffeomorphisms which are equal to a rotation near the boundary. In another point ofview, V. Humilière [22] extended the definition given by equation 3 to certain group ofcompactly supported symplectic homeomorphisms of an exact symplectic manifold p M, ω q where a compactly supported symplectic homeomorphism f of M is a C limit of a se-quence of Hamiltonian diffeomorphisms of M supported on a common compact subset of M . In the case of the open disk, for a compactly supported symplectomorphism f , thechoice of the isotopy class of f is natural. But if f is a symplectic diffeomorphism ofthe closed disk such that its restriction to the open disk is not compactly supported thenthere is no such natural choice of an isotopy from id to f .The rotation number is a well-known dynamical tool introduced by Poincaré in [31]on the group Homeo ` p S q of homeomorphisms of S which preserve the orientation. Let s consider the set of homeomorphisms r g : R Ñ R such that r g p x ` q “ r g p x q , denoted Č Homeo ` p S q . One may prove that there exists a unique r ρ P R such that for each z P R and n P Z we have | r g n p z q ´ z ´ n r ρ | ă . The number r ρ “ r ρ p r g q is called the rotationnumber of r g . Let us consider g P Homeo ` p S q and two lifts r g and r g of g in Č Homeo ` p S q ,there exists k P Z such that r g “ r g ` k and so r ρ p r g q “ r ρ p r g q ` k . Consequently we candefine a map ρ : Homeo ` p S q Ñ T such that ρ p g q “ r ρ p r g q ` Z where r g is a lift of g . Thenumber ρ p g q is called the rotation number of g . We give further details about the rotationnumber in the next section.We now state the results of this article. The following proposition allows us to considera natural choice of an action function of a symplectomorphism of the closed disk. Proposition 1.1.
Let us consider f P Diff ω p D q , A f : D Ñ R a C function such that dA f “ f ˚ λ ´ λ and µ an f invariant Borel probability measure supported on S . Then thenumber ş S A f dµ does not depend on the choice of µ and λ . The first theorem follows.
Theorem 1.1.
For each f P Diff ω p D q there exists a unique function A f : D Ñ R suchthat dA f “ f ˚ λ ´ λ and ş S A f dµ “ where λ is a Liouville form and µ a f -invariantprobability measure on S . The map Cal : Diff ω p D q Ñ R defined by Cal p f q “ ż D A f p z q ω p z q does not depend on the choice of λ and µ . Moreover the map Cal is a homogeneousquasi-morphism that extends the Calabi invariant. In another direction, the definition given by equation 3 and the definition given byequation 5 are based on isotopies. Then we consider the universal cover Ą Diff ω p D q of Diff ω p D q which is composed of couples r f “ p f, r I sq where f P Diff ω p D q and r I s is anhomotopy class of isotopies from id to f . We will prove that for f P Diff ω p D q and I anisotopy from id to f , the angle function Ang I does not depend on the choice of I P r I s .Hence, for r f “ p f, r I sq P Ą Diff ω p D q we can denote Ang r f “ Ang I for I P r I s .Moreover, for a diffeomorphism f P Diff p D q two isotopies I “ p f t q t Pr , s and I “ p f t q t Pr , s from id to f are homotopic if and only if there restriction I | S and I | S to S are homo-topics and so define the same lift Ą f | S of f | S on the universal cover over S . Hence it isequivalent to consider Ą Diff ω p D q as the set of couples r f “ p f, r φ q where f P Diff ω p D q and r φ a lift of f | S to the universal cover of S . Theorem 1.2.
Let us consider an element r f of Ą Diff ω p D q . The number Ą Cal p r f q “ ż D z ∆ Ang r f p x, y q ω p x q ω p y q , defines a morphism Ą Cal : Ą Diff ω p D q Ñ R which induces a morphism Cal : Diff ω p D q Ñ T defined for every f P Diff ω p D q by Cal p f q “ Ą Cal p r f q ` Z , where r f is a lift of f in Ą Diff ω p D q . Along the same lines, we have the following result. heorem 1.3. Let us consider an element p f, r φ q of Ą Diff ω p D q . There exists a Hamiltonianfunction p H t q t Pr , s such that H t is equal to on S for every t P R which induces an isotopy p φ t q t Pr , s from id to f where the lifted isotopy p r φ t q t Pr , s satisfies r φ “ r φ . The number Ą Cal p f, r φ q “ ż ż D H t p z q ω p z q dt, does not depend on the choice of the Hamiltonian function H . Moreover the map Ą Cal : Ą Diff ω p D q Ñ R is a morphism and induces a morphism Cal : Diff ω p D q Ñ T defined by Cal p f q “ Ą Cal p f, r φ q ` Z . Remark . We have the following commutative diagram Ą Diff ω p D q r π / / Ą Cal i (cid:15) (cid:15) Diff ω p D q Cal i (cid:15) (cid:15) R π / / T where i P t , u .The link between these three extensions is given by the following result: Theorem 1.4.
The morphisms Ą Cal and Ą Cal are equal and for r f “ p f, r φ q P Ą Diff ω p D q wehave the following equality Ą Cal p r f q “ Cal p f q ` r ρ p r φ q . Moreover the maps
Cal , Ą Cal , Cal , Ą Cal and Cal are continuous in the C topology. In the following, Ą Cal and Ą Cal will be denoted Ą Cal . Since the morphism Ą Cal and thequasi-morphism
Cal are not trivial we obtain the following corollary about the perfectnessof the groups Ą Diff ω p D q and Diff ω p D q . Recall that a group G is said to be perfect ifit is equal to its commutator subgroup r G, G s which is generated by the commutators r f, g s “ f ´ g ´ f g where f and g are elements of G Corollary 1.1.
The groups Ą Diff ω p D q and Diff ω p D q are not perfect. The non simplicity of those groups were already known since the group of compactlysupported Hamiltonian diffeomorphisms is a non trivial normal subgroup of
Diff ω p D q .The questions of the simplicity and the perfectness of groups of diffeomorphisms andHamiltonian diffeomorphism have a long story, especially the case of the group of area-preserving and compactly supported homeomorphisms of the disk D . The question appearson McDuff and Salamon’s list of open problems in [28] and we can refer for example to[3, 6, 10, 11, 27, 26, 29, 30]. Recently D. Cristofaro-Gardiner, V. Humilière, S. Seyfad-dini in [9] proved that the connected component of id in the group of area-preservinghomeomorphisms of the unit disk D is not simple. The proof requires the study of theCalabi invariant on the group of compactly supported Hamiltonian of D but also strongarguments of symplectic geometry as Embedded Contact Homology (also called ECH)developed by M. Hutchings and D. Cristofaro-Gardiner in [9].To give an illustration of the extension we compute the Calabi invariant Cal of nontrivial symplectomorphisms in sections 5 and 6. We study the Calabi invariant Cal of some irrational pseudo rotations . An irrational pseudo-rotation of the disk is an area-preserving homeomorphism f of D that fixes and that does not possess any other periodicpoint. To such a homeomorphism is associated an irrational number α R Q { Z , called the rotation number of f that measures the rotation number of every orbit around and onsequently is equal to the rotation number of the restriction of f on S . We refer to thenext section for more details.The following results of this paper are well-inspired by M. Hutchings’s recent work.M. Hutching proved as a corollary in [23] that the Calabi invariant Cal of every C irrational pseudo rotation f of the closed unit disk D such that f is equal to a rotationnear the boundary is equal to the rotation number of f . This means that for an irrationalpseudo rotation f which is equal to a rotation near the boundary, Cal p f q is equal to .The proof uses strong arguments of symplectic geometry such as the notion of open-booksintroduced by Giroux (see [18] for example) and the Embedded Contact Homology theory.We want to adopt a more dynamical point of view and we partially answer the followingquestion. Question 2.
Is the Calabi invariant
Cal p f q equal to for every C irrational pseudorotation f of D ? With the continuity of Ą Cal in the C topology, we can deduce the first result of C -rigidity as the following result. Theorem 1.5.
Let f be a C irrational pseudo rotation of D . If there exists a sequence p g n q n P N in Diff ω p D q of C diffeomorphisms of finite order which converges to f for the C topology, then Cal p f q “ . Corollary 1.2.
Let f be a C irrational pseudo rotation of D . If there exists a sequence p n k q k P N such that f n k converges to the identity in the C topology, then we have Cal p f q “ . The morphisms Ą Cal and
Cal are not continuous in the C topology, see proposition4.3. Nevertheless, by a more precise study of the definition of Cal we obtain a C -rigidityresult as follows. Theorem 1.6.
Let f be a C irrational pseudo rotation of D . If there exists a sequence p n k q k P N of integers such that p f n k q k P N converges to the identity in the C topology, thenwe have Cal p f q “ . There are already general results of C -rigidity of the pseudo-rotations. Bramhamproved [7] that every C irrational pseudo-rotation f is the limit, for the C topology, ofa sequence of periodic C diffeomorphisms. Bramham [8] also proved that if we consideran irrational pseudo-rotation f whose rotation number is super Liouville (we will definewhat it means later) then f is C -rigid. That is, there exists a sequence of iterates f n j that converges to the identity in the C -topology as n j Ñ 8 . Le Calvez [25] provedsimilar results for C irrational pseudo-rotation f whose restriction to S is C conjugateto a rotation.Then for f a C pseudo-rotation of the disk D the results of Bramham and Le Calvezprovide a sequence of periodic diffeomorphisms p g n q n P N which converges to f , the diffeo-morphism g n may not be area-preserving but let us hope to completely answer question2. In a last section we give some examples where the rotation number of a pseudo-rotationsatisfy some algebraic properties and where the hypothesis of Theorem 1.6 and Corollary1.2 are satisfied. Organization e begin to give some additional preliminaries in section . In a second section wegive the formal definitions of the Calabi invariant of equations 2, 3 and 5 and their naturalextensions given by Theorems 1.1, 1.2 and 1.3. In section we give the proof the linkbetweens these extensions given by Theorem 1.4. The last section concerns the resultsabout the computation of the Calabi invariant for pseudo rotations. Invariant measures.
Let us consider f a homeomorphism of a topological space X . ABorel probability measure µ is f -invariant if for each Borel set A we have µ p f ´ p A qq “ µ p A q . In other terms, the push forward measure f ˚ µ is equal to µ . We denote by M p f q theset of f -invariant probability measures on X . It is well-known that the set M p f q is notempty if X is compact.For a probability measure µ on D we will note Diff µ p D q the subgroup of Diff ` p D q thatis the set of orientation preserving C diffeomorphisms which preserve µ . Quasi-morphism.
A function F : G Ñ R defined on a group G is a homogeneousquasi-morphism if1. there exists a constant C ě such that for each couple f, g in G we have | F p f ˝ g q ´ F p f q ´ F p g q| ă C ,2. for each n P Z we have F p f n q “ nF p f q . Rotation numbers of homeomorphisms of the circle.
The rotation number is de-fined on the group
Homeo ` p S q of homeomorphisms of S which preserve the orientation.We begin to give the definition of the rotation number on the lifted group Č Homeo ` p S q which is the set of homeomorphisms r g : R Ñ R such that r g p x ` q “ r g p x q ` . Thereexists r ρ P R such that for each z P R and n P Z we have | r g n p z q ´ z ´ n r ρ | ă , see [24] forexample. The number r ρ is called the rotation number of r g and denoted r ρ p r g q . It defines amap r ρ : Č Homeo ` p S q Ñ R . We denote by r δ : R Ñ R the displacement function of r g where r δ p z q “ r g p z q ´ z isone-periodic and lifts where for every r g P Homeo ` p S q r ρ p r g q “ ż S δdµ “ lim n Ñ8 n n ÿ i “ δ p g i p z qq . The map r ρ is the unique homogeneous quasi-morphism from Ą Diff ` p S q to R which takesthe value on the translation by , see [17] for example. More precisely for each r f , r g P Č Homeo ` p S q it holds that | r ρ p r f q ´ r ρ p r g q| ă and for each n P Z we have r ρ p r f n q “ n r ρ p r f q .Moreover, r ρ p r g q naturally lifts a map ρ : Homeo ` p S q Ñ T . Indeed, if we consider g P Homeo ` p S q and two lifts r g and r g of g there exists k P Z such that r g “ r g hencewe have r ρ p r g q “ r ρ p r g q ` k . By the Birkhoff ergodic theorem for every z P R and every g -invariant measure µ we have r ρ p r g q “ ż S δdµ. Let us describe why r ρ is not a morphism and only a quasi-morphism. A homeomor-phism of the circle has a fixed point if and only if its rotation number is zero, see [24]chapter for more details. Below we give an example of two homeomorphisms φ and ψ f S of rotation number zero such that the composition φ ˝ ψ gives us a homeomorphismas in Figure 2 without fixed point and so the rotation number of the composition is notequal to .Let us consider the two homeomorphisms of rotation number with one fixed pointas in Figures 1 and 2. φ ψ Figure 1 φ ˝ ψ Figure 2
For g P Homeo ` p S q there is a bijection between the lifts of g to R and the isotopiesfrom id to g as follows. Let I “ p g t q t Pr , s be an isotopy from id to g , the lifted isotopy r I “ p r g q t Pr , s of I defines a unique lift r g of g . Then for an isotopy I from id to g , let usdenote r g the time-one map of the lifted isotopy r I on R , we can define the rotation number r ρ p I q P R of I to be the rotation number r ρ p r g q of r g . If we consider f a homeomorphism ofthe disk isotopic to the identity and I “ p f t q t Pr , s an isotopy from id to f then we willdenote r ρ p I | S q P R the rotation number of the restriction of the isotopy I to S . If we con-sider another isotopy I from id to g one may prove that there exists an integer k P Z suchthat I is homotopic to R k I where the isotopy R “ p R t q t Pr , s satisfies R t p z q “ z e πit forevery z P S and every t P r , s . We consider r I the lifted isotopy of I and we denote r g itstime-one map. Hence r g and r g are two lifts of g such that r g “ r g ` k and r ρ p r g q “ r ρ p r g q` k andso the number r ρ p I q does not depend on the choice of the isotopy in the homotopy class of I . Irrational pseudo rotation.
An irrational pseudo-rotation is an area-preservinghomeomorphism f of D that fixes and that does not possess any other periodic point.To such a homeomorphism is associated an irrational number α P R { Z z Q { Z , called the rotation number of f , characterized by the following : every point admits α as a rotationnumber around the origin. To be more precise, choose a lift r f of f | D zt u to the universalcovering space r D “ R ˆ p , s . There exists r α P R such that r α ` Z “ α and for everycompact set K Ă D zt u and every ǫ ą , one can find N ě such that @ n ě N, r z P π ´ p K q X r f ´ n p π ´ p K qq ñ | p p r f n p r z qq ´ p p r z q n ´ r α | ď ǫ, here π : p r, θ q ÞÑ p r cos p πθ q , r sin p πθ q is the covering projection and p : p r, θ q ÞÑ θ theprojection on the second coordinate. If moreover f is a C k diffeomorphism ď k ď `8 we will call f a C k irrational pseudo-rotation.Notice that for the rotation number α of an irrational pseudo-rotation f is equal to ρ p f | S q .One can construct irrational pseudo-rotations with the method of fast periodic ap-proximations, presented by Anosov and Katok [1]. One may see [13, 14, 15, 19, 32]for further developments about this method and see [5, 4] for other results on irrationalpseudo-rotations. In this section we will explain why the functions
Cal , Ą Cal and Ą Cal are well-defined andwe will establish the relations between them. The full statement like the continuity or thequasi-morphism property will be proved in the next section. Let us consider f P Diff ω p D q and λ a Liouville -form such that dλ “ ω . The fact that H p D , R q “ implies that the continuous -form f ˚ λ ´ λ is exact. More precisely itsintegral along each loop γ Ă D is zero. Consequently the map p r, θ q ÞÑ ş γ z f ˚ λ ´ λ is a C primitive of f ˚ λ ´ λ , equal to at the origin, where for every z P D the path γ z : r , s Ñ D is such that γ z p t q “ tz .If we suppose that f is compactly supported on ˚ D then it is natural to consider theunique C function A : D Ñ R that is zero near the boundary of D and that satisfies dA “ f ˚ λ ´ λ. (6)Without the compact support hypothesis we have the following proposition. Proposition 3.1.
If we consider a C function A : D Ñ R such that dA “ f ˚ λ ´ λ thenthe number ż S A | B D dµ does not depend on the choice of µ in M p f | S q .Proof. To prove the independence over µ there are two cases to consider. ‚ If there exists only one f | S -invariant probability measure on S the result is obvious.In this case f | S is said to be uniquely ergodic. ‚ If f | S is not uniquely ergodic then by Poincaré’s theory ρ p f | S q “ pq ` Z is rationalwith p ^ q “ . The ergodic decomposition theorem, see [24] for example, tells us thatan f | S invariant measure is the barycenter of ergodic f | S -invariant measures. Moreover,each ergodic measure of f | S is supported on a periodic orbit as follows. For z a q -periodicpoint of f | S , we define the probability measure µ z supported on the orbit of z by µ z “ q q ´ ÿ k “ δ f k p z q , here δ z is the Dirac measure on the point z P S . Hence it is sufficient to prove that ş D A p f, λ, µ z q ω does not depend of the choice of a periodic point z P S .Let us consider two periodic points z and w of f | S . We consider an oriented path γ Ă S from z to w . We compute ż S Adµ z ´ ż S Adµ w “ q q ´ ÿ k “ A p f k p z qq ´ A p f k p w qq“ q q ´ ÿ k “ ż f k p γ q dA “ q q ´ ÿ k “ ż f k p γ q f ˚ p λ q ´ λ “ ż f q p γ q λ ´ ż γ λ “ where the last equality is due to the fact that f q p γ q is a reparametrization of the path γ . Proposition 3.1 allows us to make a natural choice of the action function to define anextension of the Calabi invariant as follows. Theorem 3.1.
For each f P Diff ω p D q we consider the unique C function A f of f suchthat dA f “ f ˚ λ ´ λ and ş S A f dµ “ where λ is a Liouville form of ω and µ an f -invariantprobability measure on S . The number Cal p f q “ ż D A f p z q ω p z q does not depend on the choice of λ or µ .Proof. The independence on the measure µ comes from Proposition 3.1 and it remains toprove the independence on λ .Let us consider another primitive λ of ω . We denote A and A the two functions suchthat dA “ f ˚ λ ´ λ and dA “ f ˚ λ ´ λ and such that for each µ P M p f | S q we have ş S Adµ “ ş S A dµ “ .The -form λ ´ λ is closed because dλ ´ dλ “ ω ´ ω “ . So there exists a smoothfunction u : D Ñ R such that λ “ λ ` du . We compute dA “ f ˚ p λ ` du q ´ p λ ` du q“ f ˚ λ ´ λ ` d p u ˝ f ´ u q“ dA ` d p u ˝ f ´ u q . Thus there exists a constant c such that A “ A ` u ˝ f ´ u ` c. For a measure µ P M p f | S q the condition ş S A dµ “ “ ş S Adµ implies that ż S A dµ “ ż S Adµ ` ż Ss p u ˝ f | S ´ u q dµ ` c “ ż S Adµ,
Howeover ş S p u ˝ f | S ´ u q dµ “ since f | S preserves µ we have c “ . inally f preserves ω hence ş D p u ˝ f ´ u q ω “ and we can conclude that ż D A ω “ ż D Aω.
We compute the extension
Cal of rotations of the disk. Proposition 3.2.
For θ P R the rotation R θ of angle θ satisfies Cal p R θ q “ . Proof.
For the Liouville form λ “ r π dθ of ω we have R ˚ θ λ ´ λ “ thus the action function A is constant. So it is equal to and we obtain the result. The following interpretation is due to Fathi in his thesis [12] in the case of compactlysupported symplectic diffeomorphisms of the unit disk. This interpretation is also devel-opped by Ghys and Gambaudo in see [16].Let us consider f P Diff ` p D q and I “ p f t q t Pr , s an isotopy from id to f . For x, y P D distinct we can consider the vector v t from f t p x q to f t p y q and we denote by Ang I p x, y q the angle variation of the vector v t for t P r , s defined as follows.We have the polar coordinates p r, θ q and a differential form dθ “ udv ´ vduu ` v , where p u, v q are the cartesian coordinates. For every couple p x, y q P D z ∆ we define Ang I p x, y q “ π ż γ dθ, (7)where γ : t Ñ f t p x q ´ f t p y q .The function Ang I is continuous on the complement of the diagonal of D ˆ D . More-over, if f is at least C then the function Ang I can be extended on the diagonal into abounded function on D ˆ D . Indeed, we consider K the compact set of triplets p x, y, d q where p x, y q P D ˆ D and d a half line in R containing x and y and oriented by the vectorjoining x to y if x ‰ y . If x and y are distincts, the half line d is uniquely determined and D ˆ D z ∆ can be embedded in K as a dense and open set. We define Ang I p x, x, d q as thevariation of angle of the half lines df t p d q for t P r , s . This number is well-defined andextends Ang I into a continuous function on K .For r f “ p f, r φ q P Ą Diff ω p D q and two Hamiltonian isotopies I “ p f t q t Pr , s and I “p f t q t Pr , s from id to f associated to r φ . The isotopies I and I are homotopic so for everycouple p x, y q P D z ∆ we have ż γ dθ “ ż γ dθ, where γ : t ÞÑ f t p x q ´ f t p y q and γ : t ÞÑ f t p x q ´ f t p y q . Hence, we can define the anglefunction Ang r f of r f by Ang r f “ Ang I . We have the following lemma. emma 3.1. Let us consider r f “ p f, r φ q P Ą Diff ω p D q . For every p x, y q P D z ∆ the number Ang r f p x, y q ´ r ρ p r φ q only depends on f .Proof. Let us consider I another isotopy from id to f .There exists k P Z such that I is homotopic to R k π I and by definition of Ang h givenby equation 10 we have Ang R k π I “ Ang I ` k . Moreover I is in the same homotopyclass of R k π I and we obtain Ang I “ Ang I ` k . Since the rotation number also satisfies r ρ p I | S q “ r ρ p I | S q ` k , the result follows.Lemma 3.1 allows us to extend the Calabi invariant on the lifted group Ą Diff ω p D q asfollows. Theorem 3.2.
Let us consider r f “ p r f , r φ q P Ą Diff ω p D q . The number Ą Cal p r f q “ ż D z ∆ Ang r f p x, y q ω p x q ω p y q , defines a morphism Ą Cal : Ą Diff ω p D q Ñ R and induces a morphism on Diff ω p D q defined by Cal p f q “ Ą Cal p r f q ` Z , where r f P Ą Diff ω p D q is a lift of f .Proof. First, Ą Cal is well-defined since the angle function
Ang r f is integrable on D z ∆ .Let us consider r f “ p f, r φ q and r g “ p g, r φ q two elements of Ą Diff ω p D q and two isotopies I “ p f t q t Pr , s P r I s from id to f associated to r φ and I “ p g t q t Pr , s from id to g associatedto r φ . We consider the concatenation I ¨ I of the isotopy I and I which gives an isotopyfrom id to f ˝ g associated to r φ ˝ r φ and we define the element r f ˝ r g “ p f ˝ g, r φ ˝ r φ q P Ą Diff ω p D q .For each p x, y q P D z ∆ we have Ang I ¨ I p x, y q “ Ang I p x, y q ` Ang I p g p x q , g p y qq . Hence we obtain
Ang r f ˝ r g p x, y q “ Ang r g p x, y q ` Ang r f p g p x q , g p y qq . We integrate the previous equality and since g preserves ω we deduce that Ą Cal is a mor-phism from Ą Diff ω p D q to R .Moreover, Lemma 3.1 assures that Ą Cal induces the morphism Cal from Diff ω p D q to T .Notice that the morphisms Ą Cal and Cal satisfy the following commutative diagram Ą Diff ω p D q / / Ą Cal (cid:15) (cid:15) Diff ω p D q Cal (cid:15) (cid:15) R / / T where the horizontal arrows are the covering maps.This interpretation allows us to generalize the definition to other invariant measuresof the disk. Let us consider r f “ p f, r φ q P Ą Diff p D q and an isotopy I from id to f associated o r φ . We consider a probability measure µ on D without atom which is f -invariant. Wedefine the number r C µ p I q by r C µ p r f q “ ż ż D z ∆ Ang r f p x, y q dµ p x q dµ p y q . By Lemma 3.1 we obtain the following corollary.
Corollary 3.1.
Let us consider r f “ p f, r φ q P Ą Diff ω p D q . For every p x, y q P D z ∆ thenumber r C µ p r f q ´ r ρ p r φ q only depends on f . Birkhoff ergodic theorem gives another way to compute r C µ p r f q for r f “ p f, r φ q P Ą Diff p D q .Let us consider an isotopy I “ p f t q t Pr , s from id to f associated to r φ . For p x, y q P D ˆ D z ∆ we have Ang I n p x, y q “ Ang I p x, y q ` Ang I p f p x q , f p y qq ` ... ` Ang I p f n ´ p x q , f n ´ p x qq . (8)The function Ang I is bounded so the function y Ang I p x, y q “ lim n Ñ8 n Ang I n p x, y q , is defined µ ˆ µ almost everywhere and depends only on the homotopy class of I . Hencewe can define y Ang r f “ y Ang I . Thus we obtain the following equality r C µ p r f q “ ż ż D ˆ D y Ang r f p x, y q dµ p x q dµ p y q . (9)We state the proposition of topological invariance, see [16]. Proposition 3.3.
Let us consider two probability measures µ and µ of D without atomand two compactly supported elements of Diff µ p D q and Diff µ p D q denoted φ and φ suchthat there exists a homeomorphism h P Diff ` p D q satisfying φ “ h ˝ φ ˝ h ´ and h ˚ p µ q “ µ . We have that C µ p φ q “ C µ p φ q . For a probability measure µ of the disk, there is the equivalent result to extend theinvariant C µ . Theorem 3.3.
Let us consider an element r f P Ą Diff µ p D q . The number r C µ p r f q “ ż D z ∆ Ang r f p x, y q dµ p x q dµ p y q , defines a morphism r C µ : Ą Diff µ p D q Ñ R which induces a morphism C µ : Diff µ p D q Ñ T defined for every f P Diff µ p D q by C µ p f q “ r C µ p r f q ` Z , where r f P Ą Diff µ p D q is a lift of f . The proof of the previous theorem is basically the same as Theorem 3.2 and if weconsider the Lesbegue measure
Leb then we have r C Leb “ Ą Cal . We have the following computation in the case of the rotations. emma 3.2. For θ P R we consider r R θ “ p R θ , r r q P Ą Diff ω p D q where R θ is the rotation D Ñ D of angle θ . We have Ą Cal p r R q “ r ρ p r r q . Proof.
Let us consider R “ p R t q t Pr , s the isotopy from id to R θ given in section 2. For acouple p x, y q P D ˆ D z ∆ we consider the complex z “ x ´ y and we have for each t P r , s R t p z q “ z e itθ and we can compute Ang R p x, y q “ θ . By integration on D ˆ D z ∆ we obtain Ą Cal p r R θ q “ θ “ r ρ p r r q . In this section, the goal is to state the construction of the Calabi invariant given by equa-tion 3 in the case of compactly supported diffeomorphisms of the disk. This constructionleads to Theorem 1.3 and we explain the definition of Ą Cal given by this theorem but werefer to the next section for the proofs of certain results.Let us consider f P Diff ω p D q and a Hamiltonian isotopy I “ p f t q t Pr , s from id to f .We consider the Hamiltonian function p H t q t P R which induces the isotopy I . We denote p X t q t P R the associated vector field. We have that for every t P R , X t is tangent to S .So each H t is constant on S and we can consider p H t q t P R the associated Hamiltonianfunction such that H t | S “ . We have the following lemma.
Lemma 3.3.
The integral ż z P D ż H t p z q ω p z q dt ´ r ρ p I | S q , depends only on f .Proof. The result will be a corollary of Theorem 1.4.
Theorem 3.4.
Let us consider an element r f “ p f, r φ q P Ą Diff ω p D q and a Hamiltonianfunction H : S ˆ D Ñ R of f which induces the flow p φ t q t Pr , s such that the lift of φ | S is equal to r φ and such that H t is equal to on S for every t P R . The number Ą Cal p r f q “ ż ż D H t p z q ω p z q dt, does not depend on the choice of H . Moreover the map Ą Cal : Ą Diff ω p D q Ñ R is a morphismand Ą Cal p r f q ` Z depends only on f . It induces a morphism Cal p f q “ ż ż D H t p z q ω p z q dt ` Z , defined on Diff ω p D q . The proof comes from the equality between Ą Cal and Ą Cal which will be proven inthe next section. Moreover, the definition of Cal comes from Lemma 3.3 and we obtainthe following commutative diagram where the horizontal arrows are the universal coveringmaps. Ą Diff ω p D q / / Ą Cal (cid:15) (cid:15) Diff ω p D q Cal (cid:15) (cid:15) R / / T Proof of Theorem 1.4.
In this section, we prove Theorem 1.4.
Theorem 4.1.
The morphisms Ą Cal and Ą Cal are equal. For r f “ p f, r φ q P Ą Diff ω p D q wehave the following equality Ą Cal p r f q “ Cal p f q ` r ρ p r φ q . Moreover
Cal , Ą Cal and Ą Cal are continuous in the C topology. We separate the proof into two subsections, in the first one we establish the linksbetween the previous definitions then we prove the continuity of Ą Cal and Ą Cal . Ą Cal and Ą Cal . Proposition 4.1.
The morphisms Ą Cal and Ą Cal are equal.Proof. The proof is essentially the same as in [33], the only difference is that our sym-plectic form is normalized and the Hamiltonian diffeomorphisms that we consider is notcompactly supported in the open unit disk. Nevertheless, we verify that the proof is stillrelevant in our case.Let us consider r f “ p f, r φ q P Ą Diff ω p D q and a Hamiltonian isotopy I “ p f t q t Pr , s from id to f associated to r φ . For the proof we will give a definition of the angle function Ang I in the complex coordinates as follows. We define a -form α by α “ π d p z ´ z q z ´ z . The imaginary part satisfies dθ “ π Im p α q , where θ is the angle coordinate in the radial coordinates. For an element Z “ p z , z q P D z ∆ of we consider the curve I Z Ă D ˆ D z ∆ defined by t ÞÑ I Z p t q “ p f t p z q , f t p z qq , for each t P r , s and that for every element Z “ p z , z q P D ˆ D z ∆ p D q we have Ang I p z , z q “ π ż I Z dθ. (10)Let us consider the Hamiltonian p H t q t Pr , s which induces the flow of the isotopy I andwhich is equal to on the boundary of D . We consider the symplectic form ω “ i π dz ^ z written in the complex coordinates on D . We define ξ t “ dz p X t q and then it satisfies i X t ˆ i π dz ^ z ˙ “ i π ξ t dz ´ i π ξ t dz. By definition dH t “ ´ B H t B z dz ´ B H t B z dz, so we have ξ t “ ´ iπ B H t B z . (11) e compute the integral of the angle function ż D ˆ D z ∆ Ang I p z , z q ω p z q ω p z q “ ż D ˆ D z ∆ ż I p z ,z q π dθ ω p z q ω p z q“ Im ˜ż D ˆ D z ∆ ż I p z ,z q α ω p z q ω p z q ¸ . The following computation is well-inspired by the proof in [33]. ż D ˆ D z ∆ ż I p z ,z q α ω p z q ω p z q “ π ż D ˆ D z ∆ ż I p z ,z q d p z ´ z q z ´ z ω p z q ω p z q“ π ż D ˆ D z ∆ ż t “ ξ t p f t p z qq ´ ξ t p f t p z qq f t p z q ´ f t p z q dtω p z q ω p z q , “ π ż t “ ż D ˆ D z ∆ ξ t p f t p z qq ´ ξ t p f t p z qq f t p z q ´ f t p z q ω p z q ω p z q dt, “ ˆ π ż t “ ż z P D ż z P D zt z u ξ t p z q z ´ z ω p z q ω p z q dt “ π ż ż D ż D zt z u ´ iπ B H t B z i π dz ^ dz z ´ z ω p z q dt “ i ż ż D ż D zt z u iπ B H t B z dz ^ dz z ´ z ω p z q dt. The third equality is obtained by Fubini because the integral is absolutely integrableby Lemma 4.1. The fourth equality is due to the absolutely integrability of both terms.We established the penultimate with equation 11 and the definition of ω .We use the Cauchy formula for smooth functions (see [21]). For any C -function g : D Ñ C , we have g p w q “ iπ ż S g p z q z ´ w dz ` iπ ż D B f B z dz ^ dzz ´ w . Moreover H t is equal to zero on the boundary S and we have ż D ˆ D z ∆ ż I p z ,z q α ω p z q ω p z q “ i ż ż D H t p z q ω p z q dt. It leads to ż D ˆ D z ∆ Ang I p z , z q ω p z q ω p z q “ ż ż D H t p z q ω p z q dt. To obtain the result it remains to prove the absolute integrability we used in the compu-tation.
Lemma 4.1.
We have the following inequality ż D ˆ D z ∆ ż t “ ˇˇˇˇ ξ t p f t p z qq ´ ξ t p f t p z qq f t p z q ´ f t p z q ˇˇˇˇ ω p z q ω p z q dt ă 8 . Proof.
The total measure of D ˆ D z ∆ for ω and r , s for the Lebesgue measure is finite o by Tonnelli’s theorem it is sufficient to have the following inequalities ż t “ ż D ˆ D z ∆ ˇˇˇˇ ξ t p f t p z qq ´ ξ t p f t p z qq f t p z q ´ f t p z q ˇˇˇˇ ω p z q ω p z q dt “ ż t “ ż D ˆ D z ∆ ˇˇˇˇ ξ t p z q ´ ξ t p z q z ´ z ˇˇˇˇ ω p z q ω p z q dt ď ż t “ ż z P D | ξ t p z q| ż z P D zt z u | z ´ z | ω p z q ω p z q dt ď π ż t “ ż z P D | ξ t p z q| ω p z q dt ă 8 . To prove the second last inequality one may prove that ż z P D zt z u | z ´ z | ω p z q ď π. Remark . The number Ą Cal p f, r φ q does not depend on the choice of the isotopy in thehomotopy class of I , we obtain the same result for the construction of Ą Cal p f, r φ q whichcompletes the proof of Lemma 3.3. Proposition 4.2.
For each element r f “ p f, r φ q P Ą Diff ω p D q we have Ą Cal p r f q “ Cal p f q ` r ρ p r φ q . Proof.
Let us consider an element r f “ p f, r φ q P Ą Diff ω p D q and a Hamiltonian isotopy I “ p f t q t Pr , s from id to f associated to r φ . There exists a unique Hamiltonian function p H t q t P R which induces the isotopy I and such that H t is zero on the boundary S of D foreach t P R .We know that Cal does not depend on the choice of the primitive of ω . We considerthe Liouville -form λ “ r π dθ in the radial coordinates. We consider a probability mea-sure µ P M p f | S q .We describe the link between the action function of the first definition and the Hamil-tonian of the third definition. We consider a C family of functions p A t q t Pr , s , where A t : D Ñ R satisfies for each t P r , s dA t “ f ˚ t λ ´ λ, and such that the map A is equal to A p f, λ, µ q . So the isotopy p A t q t Pr , s satisfies d A t “ ddt p f ˚ t λ q“ f ˚ t L X t “ f ˚ t p i X t p dλ q ` d p λ p X t qqq“ d p H t ˝ f t ` λ p X t q ˝ f t q . Then there exists a constant c t : r , s Ñ R such that A t “ H t ˝ f t ` λ p X t q ˝ f t ` c t , and the map A : D Ñ R satisfies for each z P D A p z q “ ż p H t ` i X t λ qp f t p z qq dt ` ż c t dt. e denote by C the constant ş c t dt . Since the restriction of λ to S is equal to π dθ thenfor every z P S we have ż i X t λ p f t p z qq dt “ π ż dθ p BB t f t p z qq dt. Notice that the last integral is equal to the displacement function δ : R Ñ R of r φ .Moreover, the rotation number r ρ p r φ q of the isotopy I satisfies for each z P S r ρ p r φ q “ lim n Ñ8 n n ´ ÿ k “ δ p r φ k p z qq . The map z ÞÑ δ p z q is µ integrable and the Birkhoff ergodic theorem gives us ż S r ρ p r φ q dµ p z q “ ż S δ p z q dµ p z q . We obtain ż S ż i X t λ p f t p z qq dtdµ p z q “ ż S r ρ p r φ q dµ p z q “ r ρ p I | S q . Moreover, the Hamiltonian H t is equal to zero on S . So if z P S it holds that A p z q “ δ p z q ` C and consequently ż S A p z q dµ p z q “ C ` r ρ p r φ q . So the condition on A implies that C “ ´ r ρ p r φ q . Thus ż D A p z q ω p z q “ ż D ż p H t ` i X t λ qp f t p z qq dtω p z q ´ r ρ p r φ q“ ż D ż H t p f t p z qq dtω p z q ` ż D ż i X t p λ qp f t p z qq dtω p z q ´ r ρ p r φ q . We compute ş D ş i X t p λ qp f t p z qq dtω p z q . Each -form is zero on the disk so we have “ i X t p λ ^ ω q“ i X t p λ q ω ´ λ ^ i X t p ω q“ i X t p λ q ω ´ λ ^ dH t “ i X t p λ q ω ` dH t ^ λ “ i X t p λ q ω ` d p H t λ q ´ H t ω. We deduce that ż D ż i X t p λ qp f t p z qq dtω p z q “ ż D ż p H t ω ´ d p H t λ qq dt “ ż D ż H t ωdtωdt ´ ż ż S H t λdt “ ż D ż H t ωdt, here the last equality is due to the fact that f t preserves ω . Moreover H t is equal to zeroon the boundary S . We obtain ż D A p z q ω p z q “ ż D ż H t p z q ω p z q dt ´ r ρ p r φ q . We know that r ρ is a homogeneous quasi-morphism, it gives us the following corollary. Corollary 4.1.
The map
Cal : Diff ω p D q Ñ R is a homogeneous quasi-morphism.Proof. The result is straightforward because
Cal is equal to the sum of a morphism anda homogeneous quasi-morphism.Notice that Lemma 3.2 ensures that the morphisms Ą Cal (resp.
Cal ) is not zero, thenits kernel is a normal non trivial subgroup of Ą Diff ω p D q (resp. Diff ω p D q and we obtain thefollowing corollary. Corollary 4.2.
The groups Ą Diff ω p D q and Diff ω p D q are not perfect. Ą Cal . For every continuous map f from D to C we set || f || “ max x P D | f p x q| .We denote d the distance between two maps f and g of Diff p D q defined by d p f, g q “ max p|| f ´ g || , || f ´ ´ g ´ || q . We denote d the distance between two maps f and g of Diff p D q defined by d p f, g q “ max p d p f, g q , || Df ´ Dg || , || Df ´ ´ Dg ´ || q , where for every C diffeomorphism f of D , || Df || “ max x P D || D x f || .The distances d and d define naturally two distances, denoted r d and r d , on Ą Diff ω p D q defined as follows. Let us consider r f “ p f, r φ q and r g “ p g, r ψ q in Ą Diff ω p D q , we have r d p r f , r g q “ max p d p f, g q , || r φ ´ r ψ || , || r φ ´ ´ r ψ ´ || q , r d p r f , r g q “ max p d p f, g q , || r φ ´ r ψ || , || r φ ´ ´ r ψ ´ || q . We denote r id “ p id D , id R q P Ą Diff ω p D q . In this section we prove the following result. Theorem 4.2.
The map Ą Diff ω p D q Ñ R is continuous in the C topology. We need some results about the angle function.
Lemma 4.2.
Let us consider r f “ p f, r φ q P Diff ` p D q such that r d p r f , r id q ď ǫ ď { , thenfor every p x, y q P D z ∆ , it holds that | cos p π Ang r f p x, y qq ´ | ď ǫ. Proof of Lemma 4.2.
The proof is a simple computation. Let us consider x, y P D suchthat x ‰ y . One can write f “ id ` h where || h || ď ǫ and || Dh || ď ǫ . By the meantheorem we have ˇˇˇˇ h p y q ´ h p x q y ´ x ˇˇˇˇ ď ǫ. (12)We have cos p π Ang r f p x, y qq “ B f p y q ´ f p x q| f p y q ´ f p x q| ˇˇ y ´ x | y ´ x | F , here x . | . y is the canonical scalar product on R . We compute | cos p π Ang r f p x, y qq ´ | “ ˇˇˇˇB f p y q ´ f p x q| f p y q ´ f p x q| ´ y ´ x | y ´ x | ˇˇ y ´ x | y ´ x | Fˇˇˇˇ ď ˇˇˇˇ f p y q ´ f p x q| f p y q ´ f p x q| ´ y ´ x | y ´ x | ˇˇˇˇ . We compute ˇˇˇˇ f p y q ´ f p x q| f p y q ´ f p x q| ´ y ´ x | y ´ x | ˇˇˇˇ ď ˇˇˇˇ f p y q ´ f p x q ´ p y ´ x q| y ´ x | ˇˇˇˇ ` | f p y q ´ f p x q| ˇˇˇˇ | f p y q ´ f p x q| ´ | y ´ x | ˇˇˇˇ ď ˇˇˇˇ h p y q ´ h p x q| y ´ x | ˇˇˇˇ ` ˇˇˇˇ | y ´ x | ´ | f p y q ´ f p x q|| y ´ x | ˇˇˇˇ ď ˇˇˇˇ h p y q ´ h p x q y ´ x ˇˇˇˇ ď ǫ. From Lemma 4.2, we deduce the following result.
Corollary 4.3.
Let us consider r f P Ą Diff ω p D q such that d p r f , r id q ď ǫ ď { . The anglefunction satisfies || Ang r f || ď ? ǫ { π. Proof.
For every couple p x, y q P D z ∆ there exists a unique k P Z such that Ang r f p x, y q ´ k P r´ { , { q . So by Lemma 4.2 we have ě cos p| π Ang r f p x, y q ´ k |q ě ´ ǫ ě . The function arccos is decreasing so we obtain ď arccos p cos p| π Ang r f p x, y q ´ k |qq ď arccos p ´ ǫ q . Moreover the function arccos is defined on r , s and of class C on r , q such that forevery x P p , s we have p arccos p ´ x qq “ ? x ´ x ď ? x . We obtain that for every x P r , s we have arccos p ´ x q ď ? x. Hence we have π | Ang r f p x, y q ´ k | ď ? ǫ. And so | Ang r f p x, y q ´ k | ď ? ǫπ ă { . Moreover D z ∆ is path connected. Indeed, let us prove that every couple p x, y q P D z ∆ is connected to pp , q , p , qq by a path as follows. We set d the line of D passing through x and y . The line d intersects S in two points which we denote ˆ x and ˆ y such that ˆ x iscloser to x than y and ˆ y is closer to y than x as in figure 3Let us consider the path γ y : r , s Ñ D defined by γ y p t q “ t p ˆ y ´ y q ` y from y to ˆ y .The path Γ y : t Ñ p x, γ y p t qq defined on r , s sends the couple p x, y q to p x, ˆ y q . x ‚ y ‚ ˆ x ‚ ˆ y d Figure 3
Let us consider the path γ x : r , s Ñ D defined by γ x p t q “ p ´ t q x from x to p , q .The path Γ x : t Ñ p γ x p t q , ˆ y q defined on r , s sends the couple p x, ˆ y q to p , ˆ y q .Now we consider R α the rotation of D of angle α “ arg p ˆ y q . The rotation R ´ α sends ˆ y to p , q . We denote p R t q t Pr , s the isotopy from id to R α such that for every t P r , s R t is the rotation of angle tα .Hence the composition of the path Γ y , Γ x and t Ñ pp , q , R ´ t p ˆ y qq sends p x, y q to pp , q , p , qq .Moreover Ang r f is continuous on D z ∆ we deduce from the last inequality that k doesnot depend on the choice of p x, y q . The fact that r d p r f , r id q ď ǫ ď { implies that k “ and we obtain that for every p x, y q P D z ∆ | Ang r f p x, y q| ď ? ǫ { π. We now prove the continuity of
Cal for the C topology. Proof of theorem 4.2.
By Theorem 3.2 we know that Ą Cal is a group morphism. So it issufficient to prove the continuity at the identity. Let us consider r f “ p f, r φ q P Ą Diff ω p D q such that r d p r f , r id q ď ǫ ď { . By Corollary 4.3 we have for every couple p x, y q P D z ∆ | Ang r f p x, y q| ď ? ǫ { π. By integration on D z ∆ we obtain that | Ą Cal p r f q| ď ? ǫπ . Hence Ą Cal is continuous at the identity.Moreover, it is well-known that the rotation number r ρ : Č Homeo ` p S q Ñ R is continuousand we deduce from Theorem 4.1 the following corollary. Corollary 4.4.
The map
Cal : Diff ω p D q Ñ R is continuous in the C topology. Let us prove that the Calabi is not continuous in the C topology. Proposition 4.3.
The morphism Ą Cal is not continuous in the C topology. We give a counterexample which also prove that the Calabi invariant defined in theintroduction is also not continuous in the C topology, this counterexample can be findin [16] Proof.
Let us consider a sequence p h n q n ě of smooth functions h n : r , s Ñ R such that1. h n is constant near the origin,2. h n p r q is zero for r ą { n ,3. ş h n p r q πrdr “ . e consider the Hamiltonian functions H n : D Ñ R by H n p z q “ h n p| z |q . Each function H n defines a time independent vector field X n , whose induced flow is denoted φ tn . We havethe following property [16] about the computation of the Calabi invariant for compactlysupported and autonomous Hamiltonian function Proposition 4.4.
Let us consider H : D Ñ R a Hamiltonian function with compactsupport. We denote φ t the induced Hamiltonian flow and we have Cal p φ t q “ ´ πt ż D H p z q ω p z q , where Cal is the Calabi invariant defined by equation 3.
This result allows us to compute the Calabi invariant for φ n and we obtain for each n ě p φ n q “ ´ π. For each n ě we consider p φ n , id q P Ą Diff ω p D q and we have Ą Cal pp φ n , id qq “ ´ π. Moreover, φ n converges to the identity in the C topology and we obtain the result. Cal in some rigidity cases In this section, we prove several results about the Calabi invariant of irrational pseudo-rotations. C rigidity Let us begin by the simple computation of the Calabi invariant for periodic symplecticmaps.
Lemma 5.1. If f P Diff ω p D q has a finite order, then we have Cal p f q “ . Proof.
By assumption there exists p ě such that f p “ id and so Cal p f p q “ p Cal p id q “ . We deduce the following properties Proposition 5.1.
Let us consider f P Diff ω p D q . If there exists a sequence of periodicdiffeomorphisms p g k q k P N in Diff ω p D q which converges to f for the C topology, then wehave Cal p f q “ . Proof.
By Lemma 5.1 for each n P N we have Cal p g n q “ and we obtain the result bythe continuity of the map Cal for the C topology. Proposition 5.2.
Let us consider f P Diff ω p D q . If there exists a sequence p q k q k P N suchthat f q k converges to the identity in the C topology then we have Cal p f q “ . Proof.
We have
Cal p f q k q “ q k Cal p f q and Cal p f q k q converges to Cal p id q “ so Cal p f q “ . .2 C -rigidity The following theorem is a stronger version of Corollary 5.2.
Theorem 5.1.
Let us consider f P Diff ω p D q . If there exists a sequence p q k q k P N of integerssuch that p f q k q k P N converges to the identity for the C topology then we have Cal p f q “ . To prove the previous statement we will give an estimation of the angle function of f q n for a given isotopy I from id to f . For that we will consider two cases, the first oneif x is close to y and the other if x is not close to y . The following lemma gives us anevaluation of what close means. Lemma 5.2.
Let us consider f a C diffeomorphism of the unit disc D , I an isotopyfrom id to f . If d p f, id q ď ǫ ď { then for every couple p x, y q P D ˆ D which satisfies | x ´ y | ě ? ǫ, we have | cos p π Ang I p x, y qq ´ | ď ? ǫ. Proof.
Let p x, y q P D ˆ D be a couple such that | x ´ y | ě ? ǫ . Once can write f “ id ` h where h : D Ñ R satisfies || h || ď ǫ and we have ˇˇˇˇ h p y q ´ h p x q y ´ x ˇˇˇˇ ď ǫ ? ǫ “ ? ǫ. (13)We use the equation cos p π Ang I p x, y qq “ x f p y q ´ f p x q , y ´ x y| f p y q ´ f p x q| | y ´ x | (14)Moreover, if we write “ x y ´ x | y ´ x | , y ´ x | y ´ x | y we obtain cos p Ang I p x, y qq ´ “ x f p y q ´ f p x q| f p y q ´ f p x q| ´ y ´ x | y ´ x | , y ´ x | y ´ x | y (15)Equation 15 becomes ˇˇˇˇ f p y q ´ f p x q| f p y q ´ f p x q| ´ y ´ x | y ´ x | ˇˇˇˇ ď | f p y q ´ f p x q| ˇˇˇˇ | f p y q ´ f p x q| ´ | y ´ x | ˇˇˇˇ ` ˇˇˇˇ f p y q ´ f p x q ´ p y ´ x q| y ´ x | ˇˇˇˇ “ ˇˇˇˇ | y ´ x | ´ | f p y q ´ f p x q|| y ´ x | ˇˇˇˇ ` ˇˇˇˇ h p y q ´ h p x q| y ´ x | ˇˇˇˇ ď ˇˇˇˇ h p y q ´ h p x q y ´ x ˇˇˇˇ ď ? ǫ. We obtain the following lemma.
Lemma 5.3.
Under the same hypothesis, there exists an integer k P Z , uniquely defined,such that for every couple p x, y q P D ˆ D such that | x ´ y | ě ? ǫ , we have | Ang I p x, y q ´ k | ď ? ǫ { π ă { . (16) Proof.
We consider ǫ P p , { q and a couple p x, y q P D such that | y ´ x | ě ? ǫ . Bydefinition of the floor function there exists a unique k P Z such that π Ang I p x, y q ´ πk Pr´ π, π q and we have ě cos p| π Ang I p x, y q ´ πk |q ě ´ ? ǫ ě . he function arccos is decreasing so we obtain ď arccos p cos p| π Ang I p x, y q ´ πk |qq ď arccos p ´ ? ǫ q . The function arccos is defined on r , s and of class C on r , q . Moreover we have forevery x P r , q p arccos p ´ x qq “ ? x ´ x ď ? x . We obtain that for every x P r , s arccos p ´ x q ď ? x. Hence we have | π Ang I p x, y q ´ πk | ď arccos p ´ ? ǫ qď ? ǫ. Thus we have | Ang I p x, y q ´ k | ď ? ǫ { π ă { . Now we prove that k does not depend of p x, y q . Indeed the set of couples p x, y q P D such that | x ´ y | ě ? ǫ is connected in D . Indeed for a couple p x, y q P D such that | x ´ y | ě ? ǫ , let us construct a path from p x, y q to pp´ , q , p , qq .We set d the line of D passing through x and y . The line d intersects S in two pointswhich we denote ˆ x and ˆ y such that ˆ x is closer to x than y and ˆ y is closer to y than x asin the previous figure 3.Let us consider the path γ x : r , s Ñ D defined by γ x p t q “ t p ˆ x ´ x q ` x from x to ˆ x and the path γ y : r , s defines by γ y p t q “ t p ˆ y ´ y q ` y from y to ˆ y . So the path Γ : t ÞÑ p γ x p t q , γ y p t qq defined on r , s sends the couple p x, y q to p ˆ x, ˆ y q .Now we consider R α the rotation of D of angle α “ arg p ˆ x q . Notice that the rotation R ´ α sends ˆ x to p , q . We denote p R t q t Pr , s the isotopy from id to R α such that for every t P r , s R t is the rotation of angle tα . Notice that ˆ x “ ´ ˆ y and so ˆ y is send to p´ , q by R ´ α .Hence the composition of the path Γ and the path t ÞÑ p R ´ t p ˆ x q , R ´ t p ˆ y qq sends p x, y q to pp , q , p´ , qq . Moreover, ? ǫ { π ă { so k does not depend on the choice of p x, y q P D such that | x ´ y | ą ? ǫ .With these two lemmas we can give a proof of Theorem 5.1. Proof of Theorem 5.1.
We can consider I “ p f t q t Pr , s an isotopy from id to f which fixesa point of ˚ D . Up to conjugacy we can suppose that I fixes the origin and we denote I | S the restriction of I on S . We lift I | S to an isotopy p r φ t q t Pr , s on the universal coveringspace R of S such that r φ “ id and set r φ “ r φ . We will prove that Ą Cal p f, r φ q “ r ρ p r φ q andfrom Theorem 4.1 we will obtain Ą Cal p f, r φ q ´ r ρ p r φ q “ Cal p f q “ . For q P N we define the isotopy I q from id to f q as follows. We write I q “ p f qt q t Pr , s and for every z P D and t P r k ´ q , kq s we set f qt p z q “ f qt ´ k ` ˝ p f ˝ ... ˝ f q looooomooooon k ´ times . e will denote ǫ n “ d p f q n , id q . For every k P Z we can separate the difference betweenthe integral of the angle function of f q n and k into two parts as follows ş ş D ˆ D Ang I qn p x, y q ω p y q ω p x q ´ k “ ş D ´ş B ? ǫn p x q Ang I qn p x, y q ω p y q ´ k ¯ ω p x q` ş D ˆş B c ? ǫn p x q Ang I qn p x, y q ω p y q ´ k ˙ ω p x q , (17)where B c ? ǫ n p x q is the complementary of B ? ǫ n p x q in D .We can suppose that ǫ n ă { and by Lemma 5.3, there exists a unique k n P Z suchthat for each couple p x, y q P D ˆ D such that | y ´ x | ě ? ǫ n we have | Ang I qn p x, y q ´ k n | ď ? ǫ n { π. (18)Moreover, by definition there exists a sequence p ξ n q n P N of -periodic functions ξ n : R Ñ R such that || ξ || ď for every n P N and such that for every y P S and every lift r y P R of y we have Ang I qn p , y q “ r φ q n p r y q ´ r y “ q n r ρ p r φ q ` ξ n p r y q . So, for every y P S we have | Ang I qn p , y q ´ k n | “ | q n r ρ p r φ q ` ξ n p r y q ´ k n | ď ? ǫ n { π, where r y is a lift of y . Hence we obtain | q n p r ρ p r φ q ´ k n q| ď ? ǫ n { π ` . Thus we have r ρ p r φ q “ lim n Ñ8 k n q n . By equation 18 we obtain ˇˇˇˇˇż D ˜ż B c ? ǫn p x q p Ang I qn p x, y q ´ k n q ω p y q ¸ ω p x q ˇˇˇˇˇ ď ? ǫ n { π. (19)We know that for every couple p x, y q P D z ∆ and for every n P N we have Ang I qn p x, y q “ Ang I p x, y q ` Ang I p f p x q , f p y qq ` ... ` Ang I p f q n ´ p x q , f q n ´ p y qq . (20)Hence for every n P N the angle function satisfies || Ang I qn || ď q n || Ang I || . (21)We can estimate the first integral of equation 17 as follows ˇˇˇˇˇż D ˜ż B ? ǫn p x q p Ang I qn p x, y q ´ k n q ω p y q ¸ ω p x q ˇˇˇˇˇ ď ǫ n p q n || Ang I || ` | k n |q . (22)So we can deduce from the previous equations a new estimation of the Calabi invariant ˇˇˇˇż ż D ˆ D Ang I qn p x, y q ω p y q ω p x q ´ k n ˇˇˇˇ ď ? ǫ n { π ` ǫ n p q n || Ang f || ` k n q . (23)By definition we obtain ˇˇˇˇĄ Cal p f, r φ q ´ k n q n ˇˇˇˇ ď ? ǫ n q n π ` ǫ n || Ang f || ` ǫ n k n q n . (24) ence we have ˇˇˇĄ Cal p f, r φ q ´ r ρ p r φ q ˇˇˇ ď ˇˇˇˇĄ Cal p f, r φ q ´ k n q n ˇˇˇˇ ` ˇˇˇˇr ρ p r φ q ´ k n q n ˇˇˇˇ ď ? ǫ n q n π ` ǫ n || Ang f || ` q n ` ǫ n k n q n . By taking the limit on n P N , we conclude that Ą Cal p f, r φ q “ r ρ p r φ q . Remark . If we consider a sequence p r g n “ p g n , r φ n qq n P N P Ą Diff ω p D q which converges to r f “ p f, r φ q P Ą Diff ω p D q in the C topology where for each n P N , g n is a periodic diffeomor-phism of the disk and f is an irrational pseudo-rotation, then the previous method failsto prove that Ą Cal p g n , r φ n q converges to Ą Cal p f, r φ q . It is easy to see that Ang r f is close to Ang r g n but if we compute the difference Ą Cal p g n , r φ n q ´ Ą Cal p f, r φ q , as we did in equation 17,we do not have a control of || Ang r g n || so we cannot estimate properly the integral ż x ż y P B ? ǫn p x q Ang r g n p x, y q ω p x q ω p y q , where ǫ n “ || g n ´ f || . In this section, we will be interested in irrational pseudo rotations with specific rotationnumbers.
Best approximation:
Let Any irrational number α P R z Q can be written as acontinued fraction where p a i q i ě is a sequence of integers ě and a “ t α u . Conversely,any sequence p a i q i P N corresponds to a unique number α . We define two sequences p p n q n P N and p q n q n P N as follows p n “ a n p n ´ ` p n for n ě , p “ a , p “ a a ` q n “ a n q n ´ ` q n ´ for n ě , q “ , q “ a . The sequence p p n { q n q n P N is called the best approximation of α and for every n ě wehave t q n ´ α u ď t kα u , @ k ă q n where t x u is the fractional part of x P R . And for every n P N we have q n p q n ` q n ` q ď p´ q n p α ´ p n { q n , q ď q n q n ` . (25)The numbers q n are called the approximation denominators of α . .1 An example of C rigidity, the super Liouville type In this section, we show that a C irrational pseudo rotation with a super Liouville rota-tion number satisfies the assumptions of Theorem 5.1. Super Liouville.
A real number α P R { Z z Q is called super Liouville if the sequence p q n q n P N of the approximation denominators of α satisfies lim sup n q ´ n log p q n ` q “ `8 . (26)If we consider a real α P R which has super Liouville type then for each k P Z the real α ` k is also super Liouville and to simplify the notations we will say that an element r α P T is super Liouville.Bramham already showed in [8] that any C irrational pseudo-rotation f of the diskwith super Liouville rotation number is C rigid, meaning that f is the C -limit of asequence of periodic diffeomorphisms. More recently Le Calvez [25] proved that any C irrational pseudo-rotation which is C conjugated to a rotation on the boundary is C rigid. These results go as follows. Theorem 6.1.
Let us consider either a C irrational pseudo rotation or a C irrationalpseudo rotation f which is C conjugated to a rotation on the boundary.We consider α P R such that α ` Z is equal to the rotation number of f . For a sequence of rationals p p n q n q n P N which converges to α there exists a sequence p g n q n P N : D Ñ D of q n -periodic diffeomorphimsof the unit disk which converges to f for the C topology.Moreover there exists a constant C depending on f such that for every n P N we have d p f, g n q ă C p q n α ´ p n q . We deduce the following corollary.
Corollary 6.1.
Let us consider either a C irrational pseudo rotation or a C irrationalpseudo rotation f which is C conjugated to a rotation on the boundary. If the rotationnumber of f is super Liouville then we have Cal p f q “ . Proof of Corollary 6.1.
Let us consider f which is either a C irrational pseudo rotationor a C irrational pseudo rotation. We consider α P R such that α ` Z is equal to therotation number of f . We will prove that f satisfies the hypothesis of Theorem 5.1. Weconsider α P R such that α ` Z is equal to the rotation number of f and we considera sequence of rationals p p n { q n q n P N which converges to α such that q n satisfies equation25. Let p g n q n P N be the sequence of q n periodic diffeomorphisms given by Theorem 6.1associated to f and the sequence p p n { q n q n P N . We denote by K the C norm of f and weset ǫ n “ C p q n α ´ p n q { where C is the constant given by Theorem 6.1.For all k P N and each n P N the following inequality holds d p f k , g kn q ă K k ǫ n . (27)By equation 25 we can majorate ǫ n by Cq n ` to obtain for k “ q n the inequality d p f q n , id q ă K q n C p q n ` q . (28)Since K ě , equation 26 assures that lim sup n K q n p q n ` q { “ . hus we obtain that lim sup n d p f q n , id q “ . Hence up to a subsequence we can suppose that d p f q n , id q Ñ . So f satisfies the hypothesis of Theorem 5.1 and we conclude Cal p f q “ . C -rigidity, the non Bruno type Bruno type.
A number α P R z Q will be said to be Bruno type if the sequence p q n q n P N of the approximation denominators of α satisfies ÿ n “ log p q n ` q q n ă `8 . If we consider α P R which is not Bruno type then for each k P Z the real α ` k is alsonot Bruno type and to simplify the notations we will say that an element r α P T is nonBruno type.A. Avila, B. Fayad, P. Le Calvez, D. Xu and Z. Zhang proved in [2] that if we consider anumber α P R z Q which is not Bruno type, for H ą there exists a subsequence q n k of thesequence of the approximation denominators of α such that for every n P N q n j ` ě H q nj and there exists an infinite set J Ă N such that for every j P J we have t q n j α u ă e ´ qnjj . (29)We can also find the following result in the same paper. Proposition 6.1.
Let us consider a C irrational pseudo rotation f P Diff ω p D q . Supposethat ρ p f | S q is not Bruno type, then the sequence q n j satisfies d p f q nj , Id q Ñ . Hence a C irrational pseudo rotation f P Diff ω p D q satisfies the hypothesis of Corollary5.2 and we obtain the following corollary. Corollary 6.2.
Let us consider a C irrational pseudo rotation f P Diff ω p D q . Supposethat ρ p f q is not Bruno type, then we have Cal p f q “ . References [1] D. V. Anosov and A. B. Katok. New examples in smooth ergodic theory. Ergodicdiffeomorphisms.
Trudy Moskov. Mat. Obšč. , 23:3–36, 1970.[2] Artur Avila, Bassam Fayad, Patrice Le Calvez, Disheng Xu, and Zhiyuan Zhang. Onmixing diffeomorphisms of the disc.
Invent. Math. , 220(3):673–714, 2020.[3] Augustin Banyaga. Sur la structure du groupe des difféomorphismes qui préserventune forme symplectique.
Comment. Math. Helv. , 53(2):174–227, 1978.
4] F. Béguin, S. Crovisier, and F. Le Roux. Pseudo-rotations of the open annulus.
Bull.Braz. Math. Soc. (N.S.) , 37(2):275–306, 2006.[5] F. Béguin, S. Crovisier, F. Le Roux, and A. Patou. Pseudo-rotations of the closedannulus: variation on a theorem of J. Kwapisz.
Nonlinearity , 17(4):1427–1453, 2004.[6] Abed Bounemoura.
Simplicité des groupes de transformations de surfaces , volume 14of
Ensaios Matemáticos [Mathematical Surveys] . Sociedade Brasileira de Matemática,Rio de Janeiro, 2008.[7] Barney Bramham. Periodic approximations of irrational pseudo-rotations using pseu-doholomorphic curves.
Ann. of Math. (2) , 181(3):1033–1086, 2015.[8] Barney Bramham. Pseudo-rotations with sufficiently Liouvillean rotation numberare C -rigid. Invent. Math. , 199(2):561–580, 2015.[9] Dan Cristofaro-Gardiner, Vincent Humilière, and Sobhan Seyfaddini. Proof of thesimplicity conjecture, 2020.[10] Michael Entov, Leonid Polterovich, and Pierre Py. On continuity of quasimorphismsfor symplectic maps. In
Perspectives in analysis, geometry, and topology , volume296 of
Progr. Math. , pages 169–197. Birkhäuser/Springer, New York, 2012. With anappendix by Michael Khanevsky.[11] A. Fathi. Structure of the group of homeomorphisms preserving a good measure ona compact manifold.
Ann. Sci. École Norm. Sup. (4) , 13(1):45–93, 1980.[12] Albert Fathi. Transformations et homeomorphismes préservant la mesure. systèmesdynamiques minimaux.
Thèse Orsay , 1980.[13] Albert Fathi and Michael R. Herman. Existence de difféomorphismes minimaux.In
Système dynamique I - Varsovie , number 49 in Astérisque, pages 37–59. Sociétémathématique de France, 1977.[14] Bassam Fayad and Anatole Katok. Constructions in elliptic dynamics.
Ergodic TheoryDynam. Systems , 24(5):1477–1520, 2004.[15] Bassam Fayad and Maria Saprykina. Weak mixing disc and annulus diffeomorphismswith arbitrary Liouville rotation number on the boundary.
Ann. Sci. École Norm.Sup. (4) , 38(3):339–364, 2005.[16] Jean-Marc Gambaudo and Étienne Ghys. Enlacements asymptotiques.
Topology ,36(6):1355–1379, 1997.[17] Étienne Ghys.
Groups acting on the circle , volume 6 of
Monografías del Institutode Matemática y Ciencias Afines [Monographs of the Institute of Mathematics andRelated Sciences] . Instituto de Matemática y Ciencias Afines, IMCA, Lima, 1999. Apaper from the 12th Escuela Latinoamericana de Matemáticas (XII-ELAM) held inLima, June 28-July 3, 1999.[18] Emmanuel Giroux. Géométrie de contact: de la dimension trois vers les dimensionssupérieures. In
Proceedings of the International Congress of Mathematicians, Vol. II(Beijing, 2002) , pages 405–414. Higher Ed. Press, Beijing, 2002.[19] Michael Handel. A pathological area preserving C diffeomorphism of the plane. Proc. Amer. Math. Soc. , 86(1):163–168, 1982.[20] Morris W. Hirsch.
Differential topology , volume 33 of
Graduate Texts in Mathematics .Springer-Verlag, New York, 1994. Corrected reprint of the 1976 original.[21] Lars Hörmander.
An introduction to complex analysis in several variables . D. VanNostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966.[22] Vincent Humilière. The Calabi invariant for some groups of homeomorphisms.
J.Symplectic Geom. , 9(1):107–117, 2011.
23] Michael Hutchings. Mean action and the Calabi invariant.
J. Mod. Dyn. , 10:511–539,2016.[24] Anatole Katok and Boris Hasselblatt.
Introduction to the modern theory of dynamicalsystems , volume 54 of
Encyclopedia of Mathematics and its Applications . CambridgeUniversity Press, Cambridge, 1995. With a supplementary chapter by Katok andLeonardo Mendoza.[25] Patrice Le Calvez. A finite dimensional approach to Bramham’s approximation the-orem.
Ann. Inst. Fourier (Grenoble) , 66(5):2169–2202, 2016.[26] Frédéric Le Roux. Simplicity of
Homeo p D , B D , Area q and fragmentation of sym-plectic diffeomorphisms. J. Symplectic Geom. , 8(1):73–93, 2010.[27] Frédéric Le Roux. Six questions, a proposition and two pictures on Hofer distancefor Hamiltonian diffeomorphisms on surfaces. In
Symplectic topology and measurepreserving dynamical systems , volume 512 of
Contemp. Math. , pages 33–40. Amer.Math. Soc., Providence, RI, 2010.[28] Dusa McDuff and Dietmar Salamon.
Introduction to symplectic topology . OxfordGraduate Texts in Mathematics. Oxford University Press, Oxford, third edition, 2017.[29] Yong-Geun Oh. The group of Hamiltonian homeomorphisms and continuous Hamilto-nian flows. In
Symplectic topology and measure preserving dynamical systems , volume512 of
Contemp. Math. , pages 149–177. Amer. Math. Soc., Providence, RI, 2010.[30] Yong-Geun Oh and Stefan Müller. The group of Hamiltonian homeomorphisms and C -symplectic topology. J. Symplectic Geom. , 5(2):167–219, 2007.[31] Henry Poincaré.
Mémoire sur les courbes définies par une équation différentielle .Éditions Jacques Gabay, Sceaux, 1993. Reprints of the originals from 1856 through1921.[32] Frédéric Le Roux and Sobhan Seyfaddini. The anosov-katok method and pseudo-rotations in symplectic dynamics, 2020.[33] Egor Shelukhin. “Enlacements asymptotiques” revisited.
Ann. Math. Qué. ,39(2):205–208, 2015.,39(2):205–208, 2015.