A GEOMETRIC FRACTIONAL MONODROMY THEOREM

We prove the existence of fractional monodromy for two degree of freedom integrable Hamiltonian systems with one-parameter families of curled tori under certain general conditions. We describe the action coordinates of such systems near curled tori and we show how to compute fractional monodromy using the notion of the rotation number.

1. Introduction.We consider two degree of freedom integrable Hamiltonian systems defined by two Poisson commuting functions F 1 , F 2 on a symplectic manifold (P, ω).The map is called the integral map.We assume here, for simplicity, that fibres Λ f := F −1 (f ) are compact and connected.When f is a regular value of F the fibre F −1 (f ) is a smooth two dimensional torus T 2 f .We denote by R the set of regular values of F .When f is a critical value of F , i.e., when there are one or more points p ∈ F −1 (f ) where DF p is not submersive, the fibre F −1 (f ) is either a lower dimensional torus (S 1 or point) or a singular set (a singular algebraic variety when F is polynomial).
The question of the existence of global action-angle coordinates in integrable Hamiltonian systems was studied initially by Nekhoroshev [11].In [6] Duistermaat classified the obstructions to the existence of global action-angle coordinates.It turns out that an integrable Hamiltonian system on a symplectic manifold P has global action-angle coordinates if and only if the 2-torus bundle f ∈R T 2 f F R → R is trivial as a symplectic bundle [6,10,9].Here F R is the restriction of F to F −1 (R).
The most basic obstruction to the triviality of F R , and thus to the existence of global action-angle coordinates, is Hamiltonian monodromy, or simply monodromy.Monodromy characterizes the non-triviality of the bundle f ∈R H 1 (T 2 f , Z) ρ → R. The non-triviality of ρ implies the non-triviality of F R .On the other hand, the triviality of ρ does imply the existence of global action coordinates but need not imply the triviality of F R .In such case the system may still not have global angle 518 HENK BROER, KONSTANTINOS EFSTATHIOU AND OLGA LUKINA coordinates.The latter depends on the existence of a global Lagrangian section s : R → F −1 (R), for more details see [6,10,9].
The non-triviality of the bundle ρ can be uncovered by the parallel transport of a basis of H 1 (T 2 f , Z) Z 2 , f ∈ R along a closed path Γ in R that starts and ends at f .Since the fibres H 1 (T 2 f , Z) are discrete, such parallel transport is uniquely defined, see [6,10,9].Thus we obtain the monodromy map which is a group homomorphism that assigns to each homotopy class [Γ] an automorphism of H 1 (T 2 f , Z).The integrable system F in (1) has non-trivial Hamiltonian monodromy if µ in (2) is not trivial.In this case we often abuse terminology by just saying that the system has monodromy.Given a basis of H 1 (T 2 f , Z) we can write the matrix M of the automorphism µ([Γ]).M is called the monodromy matrix along Γ.Notice that an integrable system has non-trivial Hamiltonian monodromy if there is at least one closed path Γ in R such that the corresponding monodromy matrix M is not the identity.
Fractional Hamiltonian monodromy was introduced in [14] in the specific example of a two degree of freedom integrable Hamiltonian system in phase space R 4 with coordinates (q, p) = (q 1 , q 2 , p 1 , p 2 ) and standard symplectic form ω = dq 1 ∧ dp 1 + dq 2 ∧ dp 2 given by (3) F 2 commutes with J since the cubic terms of F 2 and R(q, p) = 1 2 (q 2 1 + p 2 1 ) + (q 2 2 + p 2 2 ), commute with J.The parameter ε > 0 is a scaling factor.The Hamiltonian vector field X J corresponding to J defines a locally free Hamiltonian S 1 action on R 4 * = R 4 \ {0}.On the two dimensional subset O = {(q, p) ∈ R 4 * : q 1 = p 1 = 0} this S 1 action has Z 2 isotropy.Dynamically, this is expressed by the fact that all orbits of X J in R 4 * are periodic with minimal period 2π except for those on O where the minimal period is π.
Following [7], we consider instead of F = (J, F 2 ) in (3) the integrable system E = (J, H) defined by the commuting functions Notice that the integrable systems This means that F and E define the same (up to diffeomorphism) fibration of R 4 .
In figure 1(a) we depict the bifurcation diagram BD E of E, i.e., its sets of regular and critical values.BD E contains a line C of critical values of E that joins a point c ∂ at the boundary of BD E to the origin c * = (0, 0).At the point c ∂ a Hamiltonian pitchfork bifurcation takes place.C does not include c ∂ and c * .For each critical value c ∈ C the fibre Λ c := E −1 (c) is a singular two dimensional surface that is called curled torus [14], see figure 1(b).Points at the boundary of BD E lift in phase space to smooth circles and the origin c * lifts to a degenerate singly pinched torus, see [15] for more details.The points p ∈ Λ c where rankDE p = 1 form a smooth circle o c which is an orbit of X J with period π.The set of regular values R of E (light gray in figure 1(a)) is simply connected, thus the system has trivial Hamiltonian monodromy in the sense described before.The idea presented for the first time in [14] was that in order to understand the geometry of E better we can consider the set R = R∪C which is not simply connected (since c * ∈ R) and we can ask what is the geometry of the map f ∈ R H 1 (T 2 f , Z) ρ → R. We review here the results and ideas from [14,15,7,8].
The homology group H 1 (T 2 f , Z) for f ∈ R is spanned by two homology cycles that form a basis of H 1 (T 2 f , Z).One of these cycles, denoted by b f , can be chosen as the cycle generated by the flow of X J .We denote by a f the other basis cycle of H 1 (T 2 f , Z).The homology group H 1 (Λ c , Z) of the curled torus Λ c is spanned by the cycle β c which has as representative the period-π orbit of X J and a second cycle that we denote by α c .In the limit where f → c ∈ C, the cycle b f approaches 2β c and the cycle a f approaches either α c or α c + β c depending from which side of C, f approaches c.It follows that ρ is not a fibre bundle.If we denote by H f for f ∈ R the index-2 subgroup of H 1 (T 2 f , Z) spanned by the cycles (2a f , b f ), and by H c for c ∈ C the index-4 subgroup of H 1 (Λ c , Z) spanned by (2α c , 2β c ) then the map by the parallel transport of homology cycles along closed paths Γ in R. It is necessary to consider a subgroup of the first homology group due to the fact that the only homology cycles in H 1 (T 2 f , Z) that can be continued through C are those in the subgroup H f .We come back to this point in section 3.3.For a closed path Γ that goes once around c * , see figure 1(a), there is a basis of H f in which μ([Γ]) is given by the matrix 1 0 1 1 .
The name "fractional monodromy" is due to the fact that if we write μ([Γ]) formally in a basis of H 1 (T 2 f , Z) then its matrix is 1 0 All existing proofs of fractional monodromy [15,7,8,3,16,12] are given for 1:(−2) resonant systems and use in an essential way specific properties of these systems.In this paper we give sufficient conditions so that an integrable two degree of freedom Hamiltonian system with a global S 1 action has fractional monodromy.These conditions ensure that (1) the bifurcation diagram BD of the integrable system F = (F 1 , F 2 ) has a curve of critical values C that lift to curled tori, and that (2) we can find a closed path Γ that intersects C exactly once while all other points along Γ are regular values of F .Notice that the 1:(−2) resonant systems in (3) and ( 4) have both of the above properties, thus fractional monodromy for these systems is a direct consequence of our results.Furthermore, we show how to calculate the monodromy matrix by computing the variation of the rotation number along Γ.
We give a short overview of the paper.In section 2 we state our main result, Theorem 2.1, which we prove in section 3.In section 4 we study the action coordinates of the system near the curled torus Λ c .In section 5 we show how to compute fractional monodromy through the variation of the rotation number.We close in section 6 with a discussion on the generalization of fractional monodromy to systems without a global S 1 action.
2. Statement of the main result.We consider a 2 degree of freedom integrable Hamiltonian system on a manifold P with symplectic form ω. The integral map F = (F 1 , F 2 ) : P → R 2 is assumed to be smooth and proper.Here we give conditions under which we prove that F has fractional monodromy.We distinguish between local and global conditions.2.1.Local conditions.We assume that there exists a critical value c Furthermore, we assume that (L2) critical points of F on Λ c are transversally nondegenerate, i.e., if Σ is any Poincaré section of the restriction to the F 1 = c 1 level of the Hamiltonian flow of X F1 , then the restriction of F 2 to Σ is a Morse function.This implies that F 2 is a Morse-Bott function in a neighborhood of Λ c .The local geometry of such systems was studied in [1] the results of which we use here.
Under conditions (L1) and (L2) the critical set γ c of F on Λ c consists of a finite number of periodic orbits of X F1 [1].We assume that (L3) The critical set γ c consists of exactly one periodic orbit o c of X F1 .(L4) If Σ is a two dimensional Poincaré section for o c then p = Σ ∩ o c is a saddle point for the restriction of F 2 to Σ, i.e., p is a corank-1 hyperbolic singularity of F .
Under the above conditions Λ c \ o c contains o c in its closure and is diffeomorphic either to the disjoint union of two cylinders (direct case) or to one cylinder (reverse case), see [1].In the latter case Λ c is called a curled torus.We finally assume that (L5) the critical point p is a reverse hyperbolic singularity, i.e., Λ c is a curled torus.
In this case there is a neighborhood U of c such that the set of critical values of Remark 2. Given the local conditions (L1)-(L5), the system has a local S 1 action in a neighborhood of Λ c with Z 2 isotropy on O.We discuss this point in more detail in section 3.1 but here we would like to remark that the imposed conditions exclude the cases of fractional monodromy related to higher order m:(−n) resonances [12,16] and fuzzy fractional monodromy [13].
2.2.Global conditions.We assume further that (G1) F = (F 1 , F 2 ) is invariant under a globally defined Hamiltonian S 1 action.This means that there is a function J with {J, F 1 } = {J, F 2 } = 0 such that the flow of X J is 2π periodic; the minimal period may be smaller than 2π.Condition (G1) implies that, just as in the specific example of the 1:(−2) resonant system discussed in the introduction, for each regular value f ∈ R there is a homology cycle Here C is the one dimensional set of critical values in the image of F whose points lift to a curled torus in phase space.We assume that (G2) there is a closed path such that Γ transversally crosses C exactly once for t = t * ∈ (0, 1) and Γ(t) ∈ R for t = t * .

Main result.
We can now state our main result.
Theorem 2.1.Assume that an integrable Hamiltonian system F = (F 1 , F 2 ) has a curve C of critical values c that satisfy the local conditions (L1)-(L5) and F satisfies the global conditions (G1) and (G2).Then the cycles in and there is a basis of H Γ(0) in which the monodromy matrix along the path Γ satisfying condition (G2) becomes for some k ∈ Z.
Remark 3. Expressed formally in an appropriate basis of H 1 (T 2 Γ(0) , Z), the monodromy transformation becomes thus justifying the name "fractional monodromy".3. Proof of Theorem 2.1.We look first at an integrable system F = (F 1 , F 2 ) near a critical value c that satisfies the local conditions in section 2.1.The first step of the proof in section 3.1 is the introduction near c of a local integral map (J, H) which is equivalent to F .Then, using an appropriate Poincaré section (section 3.2) we show that only a subgroup of the first homology group can be continued through C (section 3.3).We complete the proof of Theorem 2.1 in section 3.4, where we take into account the global conditions on F given in section 2.2.

3.1.
The local integral map.Let c be a critical value of the integral map F that satisfies the local conditions (L1)-(L5) in section 2.1.It follows from [1,19] that there is an open neighborhood U ⊂ image(F ) of c such that in V = F −1 (U ) there exists a unique function J (up to an additive constant) with the following properties.(J1) J commutes with F 1 and F 2 , which implies that J factors through F , i.e., J = J(F 1 , F 2 ).(J2) The flow of X J is 2π periodic outside the critical set O and π periodic in O. (J3) In O, X J = g(F 1 )X F1 , where g is a positive function.
Thus X J defines an S 1 action on V with Z 2 isotropy on O. From property (J3) of J, it follows that the map (F 1 , F 2 ) → (J, F 2 ) is a diffeomorphism in U (possibly by restricting U ). From now on we work with the integral map F local = (J, H) where Remark 4. The function J is defined locally in V .If the system has a global Hamiltonian S 1 action generated by the Hamiltonian vector field associated to a function J and the latter factors through F , then J satisfies the properties (J1)-(J3) and J − J is constant in V .
3.2.Poincaré section.Fix a critical value (j c , h c (j c )) of the integral map F local and consider the set in an open neighborhood W of p in Σ local is a smooth function t : W → R. Denote by Φ J (t, x), t ∈ R, x ∈ P , the time t flow of the Hamiltonian vector field X J and let x = Φ J (t(x), x) for x ∈ W . Then t(x ) + t(x) = 2π.We can construct a local Poincaré section Σ local with the property that the return time is π by considering the map g : W → P given by where s(x) = (t(x) − π)/2.Then points on Σ local = g(W ) return to Σ local after time π.
The structure of the level sets of H on Σ local follows from the local conditions in section 2.1.These level sets are depicted in figure 3. The intersection of Λ c with Σ local consists of the point p and the curves w s and w u that intersect transversally at p.If we consider the dynamics induced by X H when projected along the direction of X J on Σ local then the curves w s and w u can be characterized as parts of the local stable and unstable manifolds respectively of the unstable fixed point p.In order to extend this local section along Λ c we consider any smooth closed curve γ c on Λ c \ o c that connects p through a branch of w u to p through a branch of w s and such that X J is transversal to γ c .Further, denote by γ c the image of γ c under Φ J (π, −).Then γ c connects the two remaining branches of w s and w u .We represent the situation schematically in figure 4. Notice that the curves γ c and γ c can not intersect (except at p) since if they intersect at a point p , then p is a fixed point of the Poincaré return map, i.e., corresponds to an orbit of X J with Z 2 isotropy.From our assumptions the fibre Λ c contains only one such orbit.
Then we extend the local Poincaré section Σ local smoothly along γ c to obtain a surface Σ that contains γ c and such that X J is transversal to Σ .We map Σ by the symplectic map Φ J (π, −) in order to obtain a second part Σ = Φ J (π, Σ) and we define the 'global' section Σ = Σ local ∪ Σ ∪ Σ , see figure 4. Denote by λ f the intersection of the fibre Λ f = F −1 local (f ), f = (j, h) with Σ. Depending on the choice of γ c on Λ c there are two possibilities.Either for f ∈ U − , λ f consists of two components and for f ∈ U + , λ f consists of one component, or the opposite.In order to fix notation and without loss in generality we consider the first case, see figure 4. we use the Poincaré surface of section Σ constructed in section 3.2.The continuation of homology cycles through the curve C of critical values of F is also described in detail in [8].

Construction and continuation of homology cycles. For the construction of a basis of the first homology group
Consider two smooth curves ± on V ± ∩ Σ local that join (not smoothly) at p ∈ o c , see figures 3 and 4. Each curve ± intersects transversally the level sets λ f .We denote by the union of ± and we let p f = λ f ∩ and p f = Φ J (π, p f ).
The homology of the singular fibre Λ c is generated by two elements α c and β c , see figure 5(a).A representative of α c is the closed curve γ c on Σ, constructed in section 3.2 and shown also in figure 4, that connects p to itself.A representative of β c is o c .
For f = (j c , h) ∈ U − we consider the curve γ f on Σ that starts and ends at the point p f , see figure 4(a).This curve represents a homology cycle a − f , see figure 5(b).The second element of the basis of H 1 (Λ f , Z) is denoted by b − f and is represented by the X J orbit o f that starts and ends at p f .As f ∈ U − approaches c, i.e., as h → h c (j c ) − , the point p f on − approaches p.Furthermore, the representative curve γ f approaches γ c and thus a − f approaches α.As p f , f ∈ U − approaches p along − , the curve o f approaches 2o c , thus the cycle b − f approaches 2β c .For f = (j c , h) ∈ U + , λ f is a single closed curve on Λ f , see figure 4(b).Given the point p f a basis of the first homology group H 1 (Λ f , Z) is given by the cycles a + f and b + f depicted in figure 5(c).The cycle b + f is represented by the closed orbit o f of the flow of X J that starts and ends at p f .The cycle a + f is represented by the sum of the curve γ f ⊂ λ f that starts at p f and ends at p f = Φ J (π, p f ) (see figure 4(b)), and the curve o f that is defined by the X J flow and connects p f to p f .The closed curve γ f = γ f +o f represents the cycle b + f .As f ∈ U + approaches c, i.e., as h → h c (j c ) + , the point p f on + approaches p.Furthermore, the representative curve γ f approaches γ c + o c and thus a + f approaches Therefore we obtain that the limit of the basis 3.4.Fractional monodromy.In this final part of the proof we take into account the global conditions for F in section 2.2.Consider a closed curve Γ : [0, 1] → R 2 in the image of F such that Γ(t) is a regular value of F for all t ∈ [0, 1] except a t * ∈ (0, 1) for which Γ(t * ) ∈ C. Moreover we assume that Γ intersects C transversally, Γ(0) = Γ(1) = f 0 ∈ U − and that there is an > 0 such that for t * − < t < t * , Γ(t) ∈ U + , while for t Consider the basis (a − f0 , b − f0 ) of H 1 (Λ f0 , Z) defined in section 3.3.The parallel transport of this basis along Γ to a point f t = Γ(t) ∈ U + with t ∈ (t * − , t * ) gives a basis (c, d) of H 1 (Λ ft , Z).The latter basis related to the basis (a for some k ∈ Z.In general, the matrix N that describes the parallel transport from f 0 to f t is an arbitrary element of GL(2, Z). 1 Here the existence of the global S 1 action generated by the flow of X J implies that parallel transport of the cycle b − f0 ∈ H 1 (Λ f0 , Z) gives the cycle b + ft ∈ H 1 (Λ ft , Z).This determines the special form of N .
The basis (2a − f0 , b − f0 ) of the subgroup H f0 that can be continued through C, is parallel transported to (2c, d) = (2a . This means that the monodromy matrix along Γ when expressed in the basis (2a This concludes the proof of Theorem 2.1.

4.
Action coordinates near Λ c .Recall from section 3.1 that if c is a critical value of the integral map F that satisfies the local conditions in section 2.1 there is an open neighborhood V of Λ c in which we can define a local action coordinate J and a corresponding local integral map F local .Since F is an integrable Hamiltonian system a second action coordinate I exists in the preimages V ± = F −1 local (U ± ) of the simply connected sets U ± of regular values of F local .In this section we establish the relation between I in V + and I in V − .This relation is a straightforward consequence of the relation between the homology cycles a + f and a − f constructed in section 3.3.The second action coordinate I is a function in V ± which is constant on the fibres Λ f and we denote its value on Λ f by I(f ).Lemma 2. There is a second action coordinate I defined in U ± such that where f = (J(p), H(p)) and A : U → R is smooth in U \ C and continuous at C.
Proof.The symplectic form ω is exact in the open neighborhood V of Λ c (possibly by taking a subset of the original V , see Lemma 1.9 in [1]), i.e., ω = dϑ for an 1-form ϑ.This means that the second action I in V ± can be defined by integrating ϑ over any representative of the homology cycles a ± f .Recall that by construction, J is obtained by integrating ϑ over a representative of b ± f , thus where o f is defined in section 3.3.For f ∈ U let where γ f = γ f + γ f , see section 3.3, and where we used the fact that This follows from the fact that we integrate the smooth 1-form ϑ over the path γ f that in U ± depends smoothly on f .At C, A is only continuous.For f ∈ U − we define I(f ) by integrating ϑ over a representative of b − f : where we used again here that γ f ϑ = γ f ϑ.
For f ∈ U + we define I(f ) by integrating ϑ over a representative of b + f : where o f is defined in section 3.3.Then we have Thus, for f ∈ U + we obtain Remark 5.In section 3 we saw that we can consider the subgroup H f of H 1 (T 2 f , Z) which is spanned by (2a f , b f ) and the elements of which can be continued through C. At a regular fibre Λ f = T 2 f the vector fields corresponding to the action coordinates form a discrete group isomorphic to H 1 (T 2 f , Z) Z 2 .This discrete group is called the period lattice.In the case of fractional monodromy the period lattice is not defined at C because the Hamiltonian vector field X I of the second action I defined in (10) blows up at C. Any other choice of action I such that (X J , X I ) is a basis of the period lattice presents the same problem since X I = X I + nX J , for some n ∈ Z.

5.
The rotation number near Λ c and computation of fractional monodromy.Monodromy (integer or fractional) is a topological property of the fibration of the phase space defined by the integral map F .Nevertheless, monodromy can be calculated using the dynamics defined by the vector fields X F1 and X F2 in particular using the dynamical notions of first return time and rotation number.In this section we show how the fractional monodromy matrix along a closed path Γ can be computed through the total variation of the rotation number along Γ.Notice that two equivalent integral maps F and ψ • F , where ψ : R 2 → R 2 is a diffeomorphism, define the same fibration of the phase space but induce different dynamics on the invariant tori and thus define different rotation numbers.For integer monodromy, although the rotation number depends on the dynamics its variation along a path Γ depends only on the geometry of the fibration.This remains true for fractional monodromy provided that, F is chosen in such a way so that the rotation number does not diverge when Γ crosses the curve C of critical values of F so that the variation is well defined.
We give now the definition of rotation number and first return time in a system with global S 1 action generated by the flow of X J .Consider any point p on Λ f = T 2 f where f = (j, h) is a regular value of (J, H), the closed orbit o f (p) = {Φ J (t, p) : t ∈ [0, 2π]}, and the orbit of the flow of X H that starts at p and intersects o f (p) after time T (f ) at a point p .The time T (f ) is the first return time, it is independent of the point p and it is a locally smooth function of f ∈ R. Thus this construction defines a map ϕ : p → p = Φ H (T (f ), p) for all p ∈ T 2 f .As f approaches C, the first return time diverges because of the hyperbolic dynamics on Λ c , c ∈ C. The rotation number Θ(f ) ∈ [0, 2π) ⊂ R is defined as the minimum positive time that it takes to go from p to p along the flow of X J and does not depend on the choice of p ∈ Λ f .The map R → S 1 : f → [Θ(f )] is locally smooth, where [x] ∈ S 1 = R/2πZ is the equivalence class of x ∈ R with respect to the equivalence relation x ∼ x + 2kπ, k ∈ Z. Remark 6.The restriction to any X J orbit o f of the map ϕ : p → p described above is a solid rotation by angle Θ(f ) of the circle o f , see [5].
In an open neighborhood of Λ f where Θ(f ) and T (f ) are smooth define the smooth Hamiltonian vector field where f = (j, h) = (J(p), H(p)).X S is periodic with period 2π, thus S is an action coordinate.Furthermore the homology classes of the integral curves of X J and X S on the fibre Λ f form a basis of H 1 (Λ f , Z).From ( 12) we obtain S is obtained defined by integrating equation (13).
Recall that in section 4 we defined an action I such that the flows of (X J , X I ) generate a basis of H 1 (Λ f , Z).Thus we have that X S = X I + n ± X J , n ± ∈ Z for f ∈ U ± .Then from (13) we obtain that for (j, h) Using (10) we obtain that for (j, h) and Thus in order to understand the behavior of the rotation number near C we need to understand A(j, h).We have the following lemma.
Lemma 3. ∂A(j, h)/∂j is continuous at C, if and only if H| C is constant.In particular, if H| C is not constant then ∂A(j, h)/∂j blows up at C.
Proof.Assume that lim h→hc(j) ∂A(j, h)/∂j = ∂A(j, h c )/∂j.Differentiating A(j, H(j, a)) = a gives ∂H(j, a) ∂j = − ∂A(j, h) ∂h and ∂H(j, a) ∂a = ∂A(j, h) ∂h In the limit where h → h c (j) (or equivalently a → a c (j) = A(j, h c (j))), the first return time T (j, h) diverges, thus Thus DH(j, a c (j)) = 0, which implies that H is constant along the curve C parameterized here by a c (j).Now assume that h c (j) = 0 along C. For h in a compact interval [−h 0 , h 0 ] the derivative ∂A(j, h)/∂j exists uniformly in h and we obtain Lemma 3 shows that in order to compute the total variation of the rotation number along a close path Γ we should consider a Hamiltonian H with constant value H| C .This consideration played an essential role in [7].Since for an arbitrary Hamiltonian H the curve C is a graph over j, i.e., it is parameterized as (j, h c (j)) where j ∈ (j 1 , j 2 ) ⊂ R, we obtain the required Hamiltonian function H 0 by extending h c to a smooth function w over R and defining H 0 = H − w(J).Thus we have an integral map F 0 = (J, H 0 ) which is equivalent to F since the map (J, H) → (J, H 0 ) is a diffeomorphism.
In order to compute the fractional monodromy we consider the integral map F 0 and a closed path Γ : [0, 1] → R satisfying the conditions in section 2.2 and where in particular Γ(t * ) ∈ C for some t * ∈ (0, 1).Along Γ consider the function Θ : [0, 1] → R defined as the unique continuous function on [0, 1] that satisfies Θ(0) = 2Θ(Γ(0)) and Θ(t) = 2Θ(Γ(t)) mod 2π for all t ∈ [0, 1].Then the offdiagonal entry 2k − 1 in the monodromy matrix ( 8) is given by lim 6. Discussion.We have showed that in 2 degree of freedom integrable Hamiltonian systems with curled tori and a global S 1 action we have fractional monodromy.Furthermore, we showed how to compute the monodromy through the rotation number.We discuss now two further aspects of this problem: the possibility of generalizing fractional monodromy to systems without a global S 1 action, and the relation of the integer k that appears in the fractional monodromy to the type of critical values encircled by the closed path Γ. 6.1.Fractional monodromy for systems with no global S 1 action.The condition that the system has a global S 1 action plays a role only in the last part of the proof of Theorem 2.1 where we pass from the local to the global description.This condition gives the specific form of the transformation matrix N in (9).If a system does not have a global S 1 action the situation can be more complicated. 2In particular, it is possible that under a more general transformation matrix N = (n ij ) ∈ GL(2, Z), i = 1, 2, j = 1, 2, the basis (2a − f0 , b − f0 ) of H f0 is parallel transported to elements of H 1 (Λ ft , Z) that do not belong in H ft and thus can not be continued through C.This occurs when n 21 is an odd number.
A way to overcome this problem is to consider the index-4 subgroup Hf0 generated by (2a − f0 , 2b − f0 ).Notice that Hf0 is a subgroup of H f0 thus all its elements can be continued through C. The matrix N expressed in the bases (2a − f0 , 2b − f0 ) and (2a + ft , 2b + ft ) remains the same.Since N ∈ GL(2, Z) the subgroup Hf0 is mapped onto Hft generated by (2a + ft , 2b + ft ).But if then we continue the basis (2a + ft , 2b + ft ) through C we obtain (2a − f0 + b − f0 , 2b − f0 ) which spans a subgroup Hf0 of H f0 different from Hf0 .Notice that this remains true in systems with a global S 1 action.Thus we see that in the case of an integrable system with no global S 1 action we cannot define a notion of fractional monodromy that resembles the standard notion of Hamiltonian monodromy.
Remark 7. The previous discussion shows that in integrable systems with oneparameter families of curled tori, with or without a global S 1 action, we can characterize fractional monodromy as a linear transformation between two index-4 subgroups of H 1 (Λ f0 , Z).

6.2.
Critical values and the fractional monodromy matrix.The geometric monodromy theorems in [18,17,2] relate the form of the monodromy matrix along a closed path Γ to the number and type of critical values encircled by Γ.In particular, if Γ goes around k ff focus-focus points then the monodromy matrix along Γ is 1 0 k ff 1 .
In the case of fractional monodromy we found that the fractional monodromy matrix is, formally, for some k ∈ Z but we did not specify the value of k or how it is related to the type of singularities encircled by Γ.In the specific example of the 1:(−2) resonant system (4) discussed in section 1 we have = 1, cf. the fractional monodromy matrix in (6).Our conjecture is that, in this example, k = 1 is determined by the type of the degenerate singularity of F at the origin.Furthermore, the existence of a global S 1 action and the orientability of the base space of the fibration imply that we can define the sign of monodromy [4].From this we conclude that if Γ encircles the same degenerate singularity as in the 1:(−2) resonant system and also k ff focus-focus points then the off-diagonal element in the formal fractional monodromy matrix is k ff + 1 2 .The precise way on which the fractional monodromy matrix depends on the types of singularities of F encircled by Γ is one of the most interesting directions toward which this subject can be developed.6.3.Final remarks.We would like to close with a final remark and motivation for further work.Many of the constructions that were used in this paper become more natural when lifted to a covering space.Furthermore, such approach makes easier the study of higher order resonances and fuzzy fractional monodromy.We are using the notion of covering maps in order to study fractional monodromy in a forthcoming paper.

2 Figure 2 .
Figure 2. A neighborhood U of the critical value c of the integral map F that contains only critical fibres of the same type and regular fibres.The curve of critical values C separates U into two disjoint sets U + and U − .

Lemma 1 .Figure 3 .
Figure 3. Level curves of H on the local Poincaré section Σ local .

Figure 4 .
Figure 4. Intersections λ f of fibers Λ f with the Poincaré section Σ which consists of Σ local , shown here in dark gray, and the parts Σ and Σ , shown in light gray.The intersection of critical fiber Λ c with Σ is the figure-8 thin curve that consists of γ c and γ c .(a) For f ∈ U − the intersection λ f (thick curve) of the fibre Λ f with Σ consists of two closed curves γ f and γ f .The curve γ f starts and ends at the point p f ∈ − .(b) For f ∈ U + the intersection λ f (thick curve) is a single closed curve that starts and ends at the point p f ∈ + .In this case λ f consists of the open curves γ f and γ f .

Figure 5 .
Figure 5. Limits of homology cycles near the singular fibre Λ c .In this representation of the fibres the vertical lines represent orbits of X J .The lower and upper parts of these figures are identified after rotation of the upper part by π around the point p.(a) Basis(α c , β c ) of H 1 (Λ c , Z).(b) Basis (a − f , b − f ) of H 1 (Λ f , Z), f ∈ U − .(c) Basis (a + f , b + f ) of H 1 (Λ f , Z), f ∈ U + .Notice that (a − f , b − f ) → (α c , 2β c ) and (a + f , b + f ) → (α c + β c , 2β c ).
c ).The two bases (α c , 2β c ) and (α c + β c , 2β c ) span different subgroups of H 1 (Λ c , Z).This means that it is not possible to connect the homology bases (a + f , b + f ) and (a − f , b − f ) continuously through C. For this reason we consider, following [8, 7, 14, 15], the homology bases (2a − f , b − f ) and (2a + f , b + f ).As f approaches c, these two bases approach the bases (2α c , 2β c ) and (2α c + 2β c , 2β c ) respectively, that span the same index-4 subgroup H c of H 1 (Λ c , Z).Thus elements of the subgroups generated by (2a ± f , b ± f ) can be continued through the line of critical values C. We denote by H f the subgroup of H 1 for j in a compact interval [j 1 , j 2 ] we obtain that lim h→hc H(j, a) exists uniformly in j.Thus we can interchange limits to obtain lim