Coexistence of chaotic and elliptic behaviors among analytic, symplectic diffeomorphisms of any surface

We show the coexistence of chaotic behaviors (positive metric entropy) and elliptic behaviors (intregrable KAM island) among analytic, symplectic diffeomorphism of any closed surface. In particilar this solves a problem by F. Przytycki (1982).

say that it is orientation preserving and leaves Leb invariant.Then for Leb a.e. point x ∈ S, the limit Λ(x) := lim n→∞ 1. Introduction 1.1.History of the problem.-This problem enjoys a long history.The first examples of mappings with positive entropy on any surface were discovered by Katok [Kat79].These examples are isotopic to the identity.Then Katok and Gerber [GK82] obtained mappings with positive entropy on any surface in the isotopy class of any pseudo-Anosov map.Both constructions were smooth but not analytic.In [Ger85], Gerber constructed real analytic symplectic pseudo-Anosov maps on any surface, which display positive metric entropy but not the coexistence with an elliptic island.In [Prz82], Przytycki built an example of conservative diffeomorphism of the torus with coexistence of an invariant region with positive entropy and an elliptic island.His construction was infinitely smooth but not analytic.He addressed the problem of whether his construction could be generalized in the analytic class [Prz82,Rem. 1,p. 461].The issue of this problem was recalled as unclear by Liverani in [Liv04, Rem.2.4, p. 3] where a perturbation of Przytycki's example was studied.Note that Theorem A solves in particular Przytycki's problem.
In [Gor12], Gorodetski proved that typical examples of analytic symplectic surface maps are such that Λ is positive on a set of maximal Hausdorff dimension (= 2) and this coexists with elliptic islands.However this leaves open a strong version of the positive entropy conjecture which asserts that "some typical symplectic dynamics have positive metric entropy" (Λ is positive on a set of positive Lebesgue measure), see e.g.[Sin94,p. 144].A weaker version of the positive entropy conjecture proposed by Herman [Her98] asserts the existence of symplectic mappings C ∞ -close to the identity on the disk with positive metric entropy; it implies the density of surface maps with positive metric entropy among those with an elliptic cycle.In [BT19], the Herman's positive entropy conjecture was proved with Turaev.Our proof used a quotient similar to the examples of Katok and Przytycki.During Katok's memorial conference in 2019, in a conversation with Gorodetski and Kleptsyn, I claimed that the construction of [BT19] should be useful to prove the following analytic counterpart of Herman's positive entropy conjecture [Her98] and even the next analytic counterpart of our main result with Turaev.
Conjecture 1.1.-There exists an analytic and symplectic perturbation of the identity of the closed disk with positive metric entropy.
Conjecture 1.2.-For every analytic and closed symplectic surface (S, Ω), for every analytic symplectomorphism f of S displaying an elliptic periodic point, there are analytic and symplectic perturbations of f with positive metric entropy.
These conjectures 1.1 and 1.2 might be solved by translating to the analytical setting the strategy (1) of [BT19].A first step in this strategy would be to prove the analytic counterpart of Przytycki's example.Following Gorodetski this step was not in reach in a short time, and I bet with him the existence of such an example in a short time.
(2) Gorodetski offered me a nice oenological reward, as Corollaries B and C provide many examples concluding our bit: -There exists an analytic and symplectic diffeomorphism f of the closed disk displaying a stochastic island bounded by four heteroclinic bi-links which is robust relative link preservation.
Let us explain the meaning of the above statement.We recall that a stochastic island is a domain I on which the maximal Lyapunov exponent Λ is positive Leba.e.A bi-link C is a smooth circle equal to the union of two heteroclinic links C = W u (P ) ∪ {Q} = W s (Q) ∪ {P } between saddle fixed points P and Q, see Figure 4.Note that a symplectomorphism displaying a stochastic island has a positive metric entropy.Given a perturbation of the dynamics, the bi-link persists if the union of the stable and unstable manifolds of the fixed points continue to form a differentiable circle.The island is robust relative link preservation if for every C 2 -perturbation such that each of the bi-links persists, the domain bounded by the continuations of these bi-links is still a stochastic island.
We can wonder also what are the isotopy classes of analytic, symplectic surface mappings which display coexistence phenomena.Our techniques enable (at least) to obtain the following: Corollary C. -Let (S, Ω) be an analytic and closed symplectic surface and let C be an isotopy class of Diff(S).If -S is the 2-sphere or the 2-torus and C is the isotopy class of the identity, -or S is a surface of genus ⩾ 0 and C is the isotopy class of a pseudo-Anosov map of S, then there is an analytic symplectomorphism f of isotopy class C, such that f has positive metric entropy and displays elliptic islands.
(1) For an introduction to the proof of [BT19], one could look at Arnaud's Bourbaki Seminar [Arn21].
(2) More precisely the bit was that someone would prove within five years the existence of an analytic symplectomorphism of the torus, isotopic to the identity, with positive metric entropy and displaying an elliptic island.
A natural problem is: Problem 1.3.-Realize any isotopy class of surface diffeomorphisms by an analytic and symplectic dynamics displaying coexistence of positive metric entropy and elliptic islands.
It seems that the techniques of this work together with the Nielsen-Thurston's classification of symplectic dynamics on surface should lead to a solution of this problem.Another approach would be to prove Conjecture 1.2, which would imply immediately a solution to the latter problem.
The proof of Theorem A is here completely self contained.
1.2.Idea and structure of the proof.-All the proofs [Kat79, Prz82, GK82, Liv04, BT19] used bump functions to localize the surgery of the dynamics in a subset of the manifold.We recall that there is no analytic bump function.To deal with the analytic case, Gerber [Ger85] showed that the pseudo-Anosov examples of [GK82] persist in a finite co-dimensional submanifold which must intersect the (infinite-dimensional) submanifold of analytic maps.However the examples of [Prz82, Liv04, BT19], displaying the sought coexistence, persist actually along an infinite codimensional submanifold: one have to keep intact heteroclinic links.It might be possible to generalize the previous strategy by using an extension of Cartan's Theorem B. This strategy has been successfully (3) applied by Burns-Gerber [BG89] to prove that Donnay's construction [Don88] of geodesic flow on the 2-sphere with positive entropy can be performed analytically.

Instead we introduce a new approach:
We construct an analytic and symplectic extension of the surface punctured by several saddle points, so that the extended surface remains diffeomorphic to the unpunctured surface, and the analytic continuation of the dynamics on the extended surface displays elliptic islands.
We will start with an analytic, conservative dynamics with positive entropy and then we will perform blow-up, quotient, blow-down and connected sums, so that the analytic continuation of the dynamics is well defined after these operations and displays the sought coexistence properties.There is one "miracle" which enables to perform these continuations: Nearby a saddle point P , by Moser's theorem, the dynamics is the time-one map of an analytic Hamiltonian of the form P +(x, y) → H(x•y) and such can be analytically lifted to the surface blown up at P .Indeed, having a flow enables us to perform then analytic surgeries to create an integrable KAM circle which can be in turn be analytically blown down.See Figures 2  and 3.
(3) Nonetheless, the space of analytic conservative maps is more complicated to deal with than the open cone of analytic Riemannian metric.In Section 2, we present a general framework to perform these surgeries, the main novelties in these operations lie in providing sufficient conditions to obtain the analytic and symplectic continuation of a dynamic after surgery.First we recall the definition of analytic and symplectic manifolds and their mappings in Section 2.1.Then, in Section 2.2, we state Theorem 2.3 which is a general theorem used in the proof of all the surgery's results of Section 2. In Section 2.3, we present Theorem 2.5 which enables to glue two analytic and symplectic surface dynamics.In Section 2.4, we introduce Theorems 2.9 and 2.11 which allow to blow up at a hyperbolic periodic orbit of an analytic surface symplectomorphism.Finally in Section 2.5, we present Theorem 2.14 which enables to blow-down a nearby integrable circle of an analytic and symplectic surface dynamics.This last operation was perhaps the most unexpected by dynamical experts.
In Section 3, we use the surgery theorem of the previous section to construct the stochastic sphere with four holes and the integrable caps depicted in Figure 1.We start in Section 3.1 with a linear Anosov map on the 2-torus, then we blow-up four of its fixed points à la Przytycki to define an analytic symplectic diffeomorphism of the 2-torus T 2 without four disks, then we quotient it à la Katok to define an analytic symplectic diffeomorphism of the 2-sphere S without four disks in Section 3.2.These steps were already performed in [BT19] and are depicted in Figure 2.This is the construction of the stochastic spheres with four holes.Importantly nearby each component of the boundary, the dynamics is the time-one map of a Hamiltonian H on a semi-closed annulus.
Then we propose a new construction to obtain the integrable caps.In Section 3.3, we consider an analytic extension of H to a symmetric semi-closed annulus A ∆ (ε).

P. Berger
Then we perform surgeries as depicted in Figure 3. First we define a neighborhood ∆ of the circle ∂A ∆ (ε) such that ∆ has five sides, among which ∂A ∆ (ε) and two segments of orbits.We glue the two remaining sides to obtain a closed disk ∆ with two holes endowed with an analytic Hamiltonian.The borders of both holes are orbits of the systems.Thus Theorem 2.14 enables to blow-down them to obtain a closed disk ∆ endowed with an analytic Hamiltonian.The time-one map of this Hamiltonian is the integrable cap.Eventually, in Section 3.4, we show that the integrable cap recaps analytically any holes of the stochastic sphere with holes.
This allows in Section 4 to prove the main theorem and the corollaries of its proof.In Section 4.1, we start by proving Theorem A when the surface is a sphere; the construction is depicted by Figure 1.Following the number of recaped holes, coexistence phenomena are obtained on a disk (which contains the stochastic island of Corollary B), a cylinder or a pair of pants.The boundary of these can be glued together to form any closed symplectic surface, and so obtain Theorem A. A careful study enables to obtain an analytic, symplectic diffeomorphism of the torus isotopic to the identity, as wondered by Gorodetski and part of Corollary C.
In Section 4.2, we prove the remaining part of Corollary C regarding surface mappings isotopic to a pseudo-Anosov map.We will start with the example of analytic pseudo-Anosov map of [Ger85], which can represent any isotopy class of orientation preserving pseudo-Anosov maps (see also [GK82]).From this, Theorem 2.11 enables to blow-up one of its hyperbolic periodic orbit, and obtain an analytic and symplectic dynamics on the surface which is integrable nearby the holes.Then we proceed as in Section 3.3-3.4 to recap these holes.The only difference is that the normal form [Mos56] at the saddle points is more general and that we will be working on a 2-lifting of the previous construction.Caps will be replaced by a certain generalized cap given by Proposition 4.2 and Lemma 4.4.The proof of the lemma follows the same lines as Section 3.3.
Acknowledgements.-I am grateful to A. Gorodetski and V. Kleptsyn for their encouragements.I am thankful to R. Krikorian and P. Le Calvez for nice conversations.I am very grateful to S. Biebler for his careful reading.

Dynamical analytic and symplectic surgeries
In this section we revisit different surgery techniques which enable to construct a new symplectic and analytic surface from an existing one.The main novelty of this section will be to extend these operations to some symplectic and analytic dynamics on these surfaces (see Theorems 2.5, 2.9 and 2.14).These will be the basic ingredients of the proof of the main theorems; hopefully these will be also useful for other problems.

General definitions.
-To perform analytical surgeries on surfaces, we shall work with their manifold structure (rather than working on them as embedded into R 3 ).

J.É.P. -M., 2023, tome 10
We recall that an analytic manifold (resp.with boundary or with corner) M of dimension n is a paracompact space modeled on R n (resp.R + ×R n−1 or R n + ).By modeled we mean that there is an atlas of M formed by charts ϕ i : ).An analytic (or C ω ) structure is a maximal atlas.Note that the differentials of these charts form an analytic atlas (and so a C ω -structure) on the tangent space is analytic on its definition domain for every i ∈ I and j ∈ J. Then this property is satisfied with any greater C ω -atlas of M and N and in particular with the C ω -structures.
An analytic symplectic form Ω : T M ⊗2 → R on M is an analytic, bilinear, closed and non-degenerate form.Then we say that (M, Ω) is an analytic symplectic manifold.An analytic map f between two symplectic manifolds (M, Ω) and The space of analytic symplectomorphisms of (M, Ω) is denoted by Diff ω Ω (M ).A manifold is closed when M is compact and boundary less.When M is a surface with boundary or corner, we recall that it is modeled on Remark 2.1.-Given a submanifold N ⊂ M , we will denote ∂N the boundary of N as manifold and not the subset cl(N ) ∖ int(N ).These are different in general.
The above (classical) definitions may sound over formal, however they will turn out to be very efficient to verify the analyticity of the dynamics lifted by the surgeries.Also this formalism clarifies that being analytic for a mapping is a local property: In more sophisticated terms, the latter implies that the space of analytic maps defines a sheaf.x ′ = J(x).Denote π : M → M/J the canonical projection.The following will be the basis of all the surgeries performed:

A general result
Theorem 2.3.-Assume J without fixed point and satisfying the following separation criterion: Then there exists a unique C ω -manifold structure on M/J such that π : M → M/J is an analytic local diffeomorphism.Also for every analytic manifold M ′ , a map f : Proof.-It suffices to show that the following set is a closed analytic submanifold of M × M : such that pr 1 : E → M is a local diffeomorphism.Indeed [Bou67, §5.9.5] implies then the theorem.Note that Diag := {(x, x) : x ∈ M } and Graph(J) = {(x, J(x)) : x ∈ O} are submanifolds.Furthermore, they are disjoint since J does not fix any points.Furthermore, Diag is obviously closed and we can show that Graph(J) is closed.If (x n , J(x n )) n converges to some (x, y) ∈ M 2 which is not in Graph(J), then x is in cl(O) ∖ O and so in W . Thus x n ∈ W for every n large.Therefore J(x n ) is not in W by (C).So y belongs to the closed set O ∖ int(W ).Using that J is an involution, it comes that The latter proposition enables to preserve the symplectic structure: Corollary 2.4.-Under the assumptions of Theorem 2.3, if Ω is an analytic symplectic form on M such that Ω|O is left invariant by J, then there is a canonical analytic symplectic form on M/J for which π is symplectic.
; it is an involution without fixed point satisfying condition (C).Thus by the latter proposition T M ⊗2 /DJ is an analytic manifold.By uniqueness T M ⊗2 /DJ is equal to T (M/J) ⊗2 .As J leaves Ω invariant, by the last statement of Theorem 2.3, it is pushed forward by the projection T M ⊗2 → T M ⊗2 /DJ = T (M/J) ⊗2 to an analytic symplectic form on M/J.□

Symplectic gluing and induced dynamics
Theorem 2.5.-Let (M 1 , Ω) and (M 2 , Ω) be two analytic symplectic surfaces with boundary.For (2) there is an analytic symplectomorphism Then the gluing of M 1 and M 2 at C 1 and C 2 by (Φ|C 2 ) −1 •Φ|C 1 supports a structure of analytic and symplectic manifold (M, Ω) so that there exists Proof.-We are going to apply Theorem 2.3 with the symplectic manifold: its open subset: and the C ω Ω -involution J defined by: Note that Condition (C) of Theorem 2.3 is satisfied with: Hence there is a unique C ω Ω -structure on M/J such that π : M → M/J is of class C ω Ω .Note that for η ′ < η, there are canonical inclusions (M 1 ∖ C 1 ) ⊔ (M 2 ∖ C 2 ) → M/J and R/Z × (−η ′ , η ′ ) → M/J and their images form a open covering of M/J on which the maps π • f 1 , π • f 2 and π • f 12 agree.So, by Proposition 2.2, these maps define a C ω Ω -map f of M/J.As π is a local diffeomorphism, f is a local diffeomorphism.We conclude by noting that f is a homeomorphism as it is the gluing of two homeomorphisms.□ Remark 2.6.-In Theorem 2.5, we can assume that 2.4.Symplectic blow-up and induced dynamics.-Let (M, Ω) be a symplectic C ω -surface.A blow-up at a point P ∈ M ∖ ∂M consists of replacing P by a circle and a neighborhood V of P by symplectic polar coordinates.Usually these coordinates are parametrized by a Möbius strip, here we will parametrize them by a semi-closed annulus: such a blow-up.We endow the following with the standard symplectic form dx ∧ dy or dθ ∧ dr: The blow-up depends on the choice of a C ω Ω -chart φ of a neighborhood V of P of the form: φ : D(δ) −→ V ⊂ M and φ(0) = P.
Definition 2.7.-The quotient M := Å(δ) ⊔ (M ∖ {P })/J endowed with the canonical projection p : M → M is a blow-up of M at P given by φ.
Note that J has no fixed point.Also Condition (C) of Theorem 2.3 is satisfied with W = A(δ/3) ⊔ M ∖ ψ(A(2δ/3)).By Corollary 2.4, it comes: Proposition 2.8.-The space M has a canonical structure of analytic and symplectic surface.
Here is the first key ingredient of the proof of the main theorem.
Let Λ be an integral of the function λ so that Λ(0) = 0. Note that DΛ > 0 and that φ −1 •f •φ coincides with the time-one of the flow of the Hamiltonian (x, y) → Λ(x•y).Actually, in the proof of the main theorem, we will blow-up the surface at a finite set P ⊂ M .The blow-up of M at P depends on the choice of a C ω Ω -chart φ of a neighborhood V of P and of the form: We glue the surfaces A(δ) × P and M ∖ P at the open subsets Å(δ) × P and V ∖ P with the symplectomorphism: We apply Corollary 2.4, with the involution J of O = Å(δ) × P ⊔ V ∖ P which coincides with ψ on Å(δ) × P and with ψ −1 on V ∖ P, to obtain similarly: Definition 2.10.-The quotient M := M/J endowed with the canonical projection p : M → M is a blow-up of M at P given by φ.The space M has a unique analytic and symplectic surface structure such that p is analytic.

Symplectic blow-down.
-The symplectic blow-down is the inverse operation of the blow-up.This surgery will be used in Sections 3.3 and 4.2 as depicted by Figures 3 and 5   Proposition 2.13.-There exists a unique structure of analytic surface on M such that p is analytic.Moreover the C ω -symplectic form of M pushes forward to one on M for which p is symplectic.
The following states that if a symplectic diffeomorphism of a surface M is integrable and non-degenerate nearby one circle in its boundary then there is a blow-down which pushes forward the dynamics to one with an elliptic point at the blown-down circle.
Theorem 2.14.-Let ( M , Ω) be an analytic symplectic surface, let C be a circle in the boundary of M and let f ∈ Diff ω Ω ( M ) be such that its restriction to a neighborhood U of C coincides with the time-one map of the flow of a non-degenerate (4) analytic Hamiltonian H on U .Then C can be blown down by a map p : M → M and there exists f ∈ Diff ω Ω (M ) satisfying: In particular this lemma implies the existence of an analytic map h Let us perform the blow-down using the map φ : D(δ) ∖ {0} → V given by Equation (2.3) with ψ as set up by the latter lemma.This defines a surface M and a projection p : M → M .By Proposition 2.13, M has a canonical structure of symplectic and analytic surface.Note that H defines an analytic maps H on D(δ): The time-one map of the Hamiltonian H defines a C ω Ω -map on a neighborhood of 0 ∈ D(δ), whose restriction to D(δ) ∖ {0} coincides with φ −1 • f • φ|D(δ) ∖ {0}, so this defines indeed a C ω Ω -map f on M by Theorem 2.3.□

Integrable caps for stochastic spheres with four holes
In this section, we apply the surgery techniques of the previous section to construct an analytic and stochastic dynamics on the sphere without four disks and a dynamics on a disk which enables to recap analytically these holes.We start with the Anosov map A(x, y) = (13 • x + 8 • y, 8 • x + 5 • y) which acts on the torus T 2 := R 2 /Z 2 endowed with the symplectic form Ω = dx ∧ dy.Let R ∈ O 2 (R) and λ > 0 be such that A = R −1 × diag(exp(λ), exp(−λ)) × R. The set P := {0, (1/2, 0), (0, 1/2), (1/2, 1/2)} is formed by four fixed points of the Anosov map A.
J.É.P. -M., 2023, tome 10 For δ > 0 sufficiently small, the following is a diffeomorphism onto a neighborhood V of P: Observe that φ −1 • A • φ coincides with the time-one of the flow of H : (x, y, P ) ∈ Then T 2 is given by gluing A(δ) × P and T 2 ∖ P at the open subsets Å(δ) × P and V ∖ P with Hence with ψ : A(δ) × P → T 2 the canonical inclusion onto a neighborhood V := p −1 (V ) of the boundary of T 2 , we have that ψ −1 • A • ψ coincides with the time-one map of the Hamiltonian:

3.2.
A stochastic dynamics on the sphere with four holes.-In this subsection we are going to construct an analytic and symplectic non-uniformly hyperbolic dynamics g of the sphere with fours holes S as in Figure 1.In order to do so, we proceed as depicted in Figure 2 [center-right], by taking the quotient of T 2 by an involution Γ that we shall define.We recall that T 2 is the quotient of the disjoint union of A(δ) × P with T 2 ∖ P and the involution induced by the map ψ of Equation (3.1).We identity A(δ) × P and T 2 ∖ P to open subsets of T 2 , using the projection p whose restriction to each latter set is an embedding.Recall that in this identification, A acts on T 2 ∖ P as A and its restriction to A(δ) × P coincides with the time-one map of the Hamiltonian H(θ, r, P ) = λ • r • sin(2θ).
The involution − id on T 2 fixes each point of P and lifts to T 2 as the involution Γ whose restriction to T 2 ∖ P is equal to − id and whose restriction to A(δ) × P is Note that Γ is an analytic symplectomorphism which leaves invariant the subsets T 2 ∖ P and A(δ) × P of T 2 and acts freely on them.Thus π Γ := T 2 → T 2 /Γ is a 2-covering.Observe that S := T 2 /Γ is a sphere without four holes.As A•(− id) = −A and H(θ + π, r, P ) = H(θ, r, P ), we have Using Theorem 2.3 and Corollary 2.4 with O = M = S and J = Γ, it comes that S = T 2 /Γ has a canonical structure of symplectic and analytic surface for which the 2-covering π Γ is symplectic and analytic.Moreover the dynamics A descends to an analytic and symplectic dynamics g on S. In other words, there is g ∈ Diff ω Ω ( S) such that: As the hyperbolic map A|T 2 ∖ P is a lifting of g|S ∖ ∂S, the map g has positive metric entropy.Let us now describe the dynamics of g at the neighborhood of ∂S.Let A Γ (δ) := R/πZ × [0, δ) J.É.P. -M., 2023, tome 10 and note that A Γ (δ) × P is equal to the quotient A(δ) × P/Γ.Denote also by H the analytic function such that H = H • π Γ , which is: Note that there is a C ω -symplectomorphism ψ Γ from A Γ (δ) × P onto the neighborhood coincides with the time-one map of the flow of H.To summary we obtained: Claim 3.1.-There is a symplectic sphere with four holes ( S, Ω) and g ∈ Diff ω Ω ( S) such that: (1) every point x ∈ S has positive Lyapunov exponent: lim sup 1 n log ∥Dg n ∥ → ∞, (2) there are δ > 0 and a C ω -symplectomorphism ψ Γ from A Γ (δ)×P onto a neighborhood of ∂ S ⊂ S, such that g coincides with the time-one map of the Hamiltonian flow of -We are now going to construct the cap which recaps the holes of S. For ε > 0, let ).We will see in the next subsection that the following claim provides the sought cap: Claim 3.2.-There exist a C ω -Hamiltonian H ∆ on a closed symplectic disk (∆, Ω) which satisfies: (1) H ∆ has only two critical points in ∆ ∖ ∂∆, the Hessian is definite positive at them, (2) there are ε ∈ (0, 1 2 δ 2 ) and a In this subsection we show this claim, by proceeding as in Figure 3. First we shall define ∆ ⊂ A ∆ (δ) as in Figure 3 [left].For a small ε ∈ (0, δ) fixed later, we put: Now fix ε > 0 small enough so that the set cl( ∆) is a pentagon whose sides are R/πZ × {0}, L + , L − , Σ in and Σ out , where: ) is tangent to the sides of R/Z × {0}, L + and L − .On the other hand, it enters into ∆ by Σ in and exits ∆ by Σ out .Indeed, we have:

ε}, and
and on these segments, the r-component of ∂ t ϕ t (π/2, r)| t=0 is equivalent (as ε is small) to resp.2λ • ε and −2λ • ε.As in Figure 3 [left], we glue ∆ to itself at: and vice-versa for every t ∈ (0, 1).Note that we can use Theorem 2.3 with M = ∆ and O := ∆ in ⊔ ∆ out since condition (C) is satisfied with the following neighborhood of cl(O) ∖ O ⊂ ∆: where Using that J is symplectic and leaves H equivariant (H • J|O = H|O), Theorem 2.3 and Corollary 2.4 asserts that the quotient ∆ := ∆/J has a unique structure of C ω Ω -surface for which π J : ∆ → ∆ is of class C ω Ω and for which there exists H ∈ C ω ( ∆, R) satisfying: We notice that ∆ is symplectomorphic to a closed disk D without two open disks D + and D − , as depicted in Figure 3 [center]: We chose the identification such that the circle ∂D ± is the quotient of L ± for each ± ∈ {−, +} and the circle ∂D is the quotient of R/πZ × {0}.
As the symplectic gradient of H is colinear to R/πZ × {0} and L ± , and moreover non-degenerate at ∆ ∖ R/πZ × {0}, the same occurs for the quotient: the symplectic gradient of H is colinear to the boundary ∂ ∆ and non-degenerate on D ∖ (D + ⊔ D − ).
Hence we can apply Theorem 2.14 twice to blow down each of the holes of ∆.This defines a symplectic closed disk ∆ and a C ω -map p : ∆ → ∆ so that p sends ∂D + ⊔ ∂D − to two points {p + , p − } ⊂ ∆ and the restriction p| ∆ ∖ (∂D + ⊔ ∂D − ) is a symplectomorphism onto ∆ ∖ {p + , p − }.Moreover, Theorem 2.14 implies that there is a C ω -Hamiltonian H ∆ on ∆ such that H ∆ • p = H for which p + and p − are elliptic.As H has no critical point, it comes that H ∆ has no critical point on int ∆∖{p − , p + }.This gives the first statement of Claim 3.2.The second statement is obvious since p is a symplectomorphism from a neighborhood of ∂D ⊂ ∆ onto a neighborhood of ∂∆ ⊂ ∆, and since π J is a symplectomorphism from a neighborhood of ∂∆ ⊂ ∆ onto a neighborhood of R/πZ × {0} in ∆.Hence for ε > 0 sufficiently small, the restriction 3.4.Gluing the cap ∆ to a hole of S. -In this subsection we show the following: Claim 3.3.-For every 1 ⩽ n ⩽ 4, the symplectic and analytic surface ( S, Ω) can be extended to a symplectic and analytic surface (M, Ω) which is the union of S and n-copies of the disk ∆, each of which is glued at its boundary to a different component of ∂ S, and such that there is a C ω -symplectomorphism f M of M whose restriction to S is g and whose restriction to each copy of ∆ is the time-one map f of the Hamiltonian H ∆ .
Proof.-Let us show the case n = 1.Let C be a component of ∂ S.
Finally note that the case 4 ⩾ n > 1 can be proved by induction on n using the later argument of the inductive step.□ Then each of the four holes of S are recapped with a copy of the disk ∆, so that M is a symplectic sphere S. The claim asserts the existence of an analytic symplectomorphism f S whose restriction to S ⊂ S is the stochastic map g and whose restriction to the complement S ∖ S is equal to four copies of the cap dynamics h which displays each time two elliptic islands and so eight in total.

Application of the construction
Proof of Theorem A. Case where S is the torus.-We apply Claim 3.3 with n = 2. Then two holes of S are recapped with two copies of the disk ∆, so that M is an annulus A. The claim asserts the existence of an analytic symplectomorphism f A whose restriction to S ⊂ A is the stochastic map g and whose restriction to the So it suffices to glue the two boundaries of ∂A so that the quotiented dynamics remains analytic (and symplectic).To this end, we apply Theorem 2.5 with and the map Φ : ψ(A(δ) × {+1, −1}) → R/πZ × (−δ, δ) which sends ψ(θ, r, ±1) to (±θ, ±r) for every (θ, r, ±1) ∈ A(δ) × {+1, −1}.
Proof of Theorem A. Case where S is a surface of higher genus.-We apply Claim 3.3 with n = 1.Then M is a pair of pants P: a disk with two holes.The dynamics f P on P is of class C ω Ω and is stochastic at S ⊂ P and integrable at one cap ∆ with exactly two elliptic islands.We recall that every closed, oriented surface S of genus ⩾ 2 displays a pants decomposition.We glue canonically (using Theorem 2.5 as above) the pants at their boundaries to obtain the sought dynamics.□ Proof of Corollary C for S equal to the torus and f isotopic to the identity We constructed above a symplectic and analytic map f A on the closed annulus A satisfying the coexistence phenomena and moreover the following property.There is an open neighborhood N of the boundary ∂A which is symplectomorphic to A(δ) × {+1, −1}, via a C ω -map ψ which conjugates the dynamics f A |N to the Hamiltonian flow of H : (θ, r, ±1) → λ • r • sin(2θ).
In the proof of Theorem A, we glued the two components C + and C − of ∂A to obtain a dynamics on the torus displaying the coexistence phenomena.Nevertheless this dynamics is a priori in a non-trivial isotopy class.To vanish this isotopy class, the idea is to glue f A with its inverse (f A ) −1 .To this end, let f 1 be the dynamics induced by f A on the copy M 1 = A × {1} of A, and let f 2 be the dynamics induced by (f A ) −1 on another copy M 2 = A × {−1} of A.
At the boundary C + ⊔C − of A, the map (f A ) −1 is conjugated via ψ to the time-one map of the flow of −H(θ, r, ±1) = H(−θ −π/2, −r, ±1).So we can apply Theorem 2.5 to glue M 1 and M 2 at C + × {1} and C + × {−1} with the following map: Similarly the gluing is done at C − × {1} and C − × {−1} with the following map: Then observe that the surface obtained after these two gluings is a symplectic torus endowed with a C ω Ω -dynamics f whose restriction to the halve of this torus is f A and to other other halve is (f A ) −1 .Hence f displays the coexistence phenomena and is isotopic to the identity (as the twist of (f A ) −1 vanishes the one of f A ). □ called a stochastic island.This means that I is a disk with three holes; and that the boundary of I is formed by four pairs of heteroclinic bi-links {( Ľa i , Ľb i ) : 0 ⩽ i ⩽ 3}.Each Ľa i ∪ Ľb i is a smooth circle included in the stable and unstable manifolds of hyperbolic fixed points Pi and Qi respectively: For every f which is C 1 -close to f D , for every 0 ⩽ i ⩽ 3, the hyperbolic continuations P i and Q i of Pi and Qi are uniquely defined hyperbolic fixed points for f .If -For every conservative map f which is C 2 -close to f D if the bi-links are persistent, then the continuations of these bi-links bound a stochastic island.In particular, the metric entropy of f is positive.

Proof of Corollary
C. -To achieve the proof of Corollary C, it remains the case of mappings isotopic to pseudo-Anosov maps (the case of the torus has been done above and the case of the sphere is an immediate consequence of Theorem A).
To carry them we use the following generalization of the cap's construction: Proposition 4.2.-Let (S, Ω) be a symplectic surface and let f ∈ Diff ω Ω (S) be displaying a periodic hyperbolic orbit with positive eigenvalues.Let ( S, ω) → (S, Ω) be the blow up given by Theorem 2.11 and let f ∈ Diff ω Ω ( M ) be the lifting of f .Then there is an analytic extension ( S, Ω) ⊃ ( S, Ω) and an extension f ∈ Diff ω Ω ( S) of f such that S is diffeomorphic to S and S ∖ S consists of a finite union of disks on which f is the product of a cycle k ∈ Z/nZ → k + 1 with an integrable map of the disk displaying three elliptic fixed points.This proposition is proved below.
Proof of Corollary C for f isotopic to a pseudo-Anosov map.. -Let (S, Ω) be a symplectic orientable, closed surface.Then by [GK82,Ger85], any orientation preserving pseudo-Anosov isotopy class is represented by an analytic symplectomorphism f .Then observe that Corollary C follows immediately from Proposition 4.2 and the next lemma.□ Lemma 4.3.-The map f displays a hyperbolic periodic cycle (P i ) i∈Zq with positive eigenvalues.
Proof.-As f has positive topological entropy, it displays a horseshoe [Kat80] with at least two rectangles.There are two possibilities: Either one of these rectangles is not rotated by the induced dynamics, and so we get immediately a saddle periodic cycle with positive eigenvalues.Or both rectangle are rotated by a half turn.Then we can compose the induced dynamics by these two rectangles to obtain a hyperbolic periodic orbit with positive eigenvalues.□ Proof of Proposition 4.2.-Let P be the periodic orbit which is blown up and let p : S → S be the canonical projection.By Theorem 2.11, there are coordinates ψ : A(δ) × P → V of a neighborhood V of p −1 (P) and a Hamiltonian H ∈ C ω (A(δ) × P, R) whose time-one map ϕ 1 H satisfies: f • ψ(θ, r, P ) = ψ(ϕ 1 H (θ, r, P ), f (P )) and H(r, θ, P ) = Λ(r • sin(2θ)) for every (θ, r, P ) nearby R/2πZ × {0} × P and for a function Λ ∈ C ω ([0, 1 2 δ 2 ), R) with positive first derivative.Note that Λ does not depend on P ∈ P because P is formed by a unique orbit.Hence on V the dynamics is conjugated to the product of the shift map on P with the time-one map f o of the Hamiltonian flow of: Observe that on C = R/2πZ × {0}, the flow of H o displays four saddle fixed points Q q = ((q/2 + 1)π/4, 0) with q ∈ Z/4Z, so that In particular C consists of four heteroclinic links.Note also that H o can be canonically extended to R/2πZ × (− 1 2 δ 2 , 1 2 δ 2 ).Thus we can use the next lemma with k = 2 to recap each hole of S (as we did in Claim 3.3) of S and obtain the sought surface and dynamics (more precisely the extension is ϕ t (Σ in ) ⊂ ∆, using the C ω Ω -involution J which swap for every t ∈ (0, 1) and k ∈ Z k , each pair of points ϕ −t (θ 0 , −ε) and ϕ 1−t (θ ′ 0 , −ε) among (θ 0 , −ε) ∈ Σ out,k and (θ ′ 0 , −ε) ∈ Σ in,k such that H o (θ 0 , −ε) = H o (θ ′ 0 , −ε).Then Theorem 2.3 and Corollary 2.4 assert that the quotient ∆ := ∆/J has a unique structure of C ω Ω -surface for which π J : ∆ → ∆ is of class C ω Ω .Moreover as J leaves H o equivariant, there exists H ∈ C ω ( ∆, R) satisfying: We notice that ∆ is equal to the closed disk D without k + 1 disks (D i ) 0⩽i⩽k as depicted in Figure 5 [center]: Also on ∂D i , the Hamiltonian H is equal to ε 2 or −ε 2 .Hence the symplectic gradient of H is colinear to each boundary ∂D i .Moreover its symplectic gradient does not display critical point at these circles.So we can blow down each of the k + 1-holes D i using Theorem 2.14 as depicted in Figure 5 [center-right].These blow-downs define a symplectic closed disk (∆, Ω) endowed with an analytic Hamiltonian H ∆ satisfying the second item of the lemma.As the unique critical points of H o | ∆ were (Q i ) i∈Z 2k , these surgeries create only k + 1-new critical points at P i which are all with definite positive Hessian.□

Figure 1 .
Figure1.Analytic and conservative dynamics on a sphere displaying coexistence of a stochastic region with elliptic islands.
for analytic and symplectic surgeries.-Let M be a C ωmanifold possibly with boundaries and possibly not connected.Let O be an open subset of M and let J ∈ Diff ω (O) be an analytic involution: J 2 = id.Note that J must preserve the boundary of M : J(O ∩∂M ) = J(O)∩∂M .Let M/J be the quotient of M by the equivalence relation defined by x ∼ x ′ if either x = x ′ , or x ∈ O and J.É.P. -M., 2023, tome 10 Hence the time-one map ϕ 1 H of the flow of the Hamiltonian H : (θ, r) ∈ A(δ) −→ Λ(r • sin(2θ)) coincides with ψ −1 • f • ψ on the intersection of their definition domains.Thus by Theorem 2.3, the maps f |M ∖ {P } and ϕ 1 H define a map f ∈ Diff ω Ω ( M ) which satisfies the sought properties.□ [center-right].Let ( M , Ω) be a symplectic surface with boundary ∂ M .Assume that a component of ∂ M is a circle C. Let: We glue the surfaces D(δ) and M ∖ C at the open subsets D(δ) ∖ {0} and U ∖ C with the diffeomorphism φ.To apply Theorem 2.3, we define the involution J of O = D(δ) ∖ {0} ⊔ U ∖ C which coincides with φ on D(δ) ∖ {0} and with φ −1 on U ∖ C. Note that Condition (C) of Theorem 2.3 is satisfied with W := D(δ/2)⊔ M ∖ φ(D(δ/2)).Definition 2.12.-The quotient M := D(δ) ⊔ ( M ∖ C)/ J is called a blow-down of M at C. Denote by p : M → M the canonical projection.By Theorem 2.3 and Corollary 2.4 it comes the following: 3.1.A stochastic dynamics on the torus without four disks.-This step is depicted in Figure 2 [left-center].

Figure 2 .
Figure 2. Surgery on an Anosov map

Figure 3 .
Figure 3. Making an integrable cap by gluing the green rectangles together and then blowing down.

4. 1 .
Proof of Theorem A and Corollary B Proof of Theorem A. Case where S is the sphere.-We apply Claim 3.3 with n = 4.

J
.É.P. -M., 2023, tome 10 complement A ∖ S is equal to two copies of the cap dynamics f which displays each time two elliptic islands and so four in total.Moreover, there is an open neighborhood N of the two circles ∂A which is symplectomorphic to A(δ) × {+1, −1}, via a C ω -symplectomorphism ψ which conjugates the dynamics f A |N to the time-one map of the Hamiltonian H : (θ, r, ±1) → λ • r • sin(2θ).
then we say that the bi-links are persistent for the perturbation f .Then the next proposition implies Corollary B. □ Proposition 4.1 ([BT19, Prop.2.1]).