Monodromy Map Research Articles

The topology of the monodromy map of a second order ODE

We consider the following question: given A ∈ SL ( 2 , R ) , which potentials q for the second order Sturm–Liouville problem have A as its Floquet multiplier? More precisely, define the monodromy map μ taking a potential q ∈ L 2 ( [ 0 , 2 π ] ) to μ ( q ) = Φ ˜ ( 2 π ) , the lift to the universal cover G = SL ( 2 , R ) ˜ of SL ( 2 , R ) of the fundamental matrix map Φ : [ 0 , 2 π ] → SL ( 2 , R ) , Φ ( 0 ) = I , Φ ′ ( t ) = ( 0 1 q ( t ) 0 ) Φ ( t ) . Let H be the real infinite-dimensional separable Hilbert space: we present an explicit diffeomorphism Ψ : G 0 × H → H 0 ( [ 0 , 2 π ] ) such that the composition μ ○ Ψ is the projection on the first coordinate, where G 0 is an explicitly given open subset of G diffeomorphic to R 3 . The key ingredient is the correspondence between potentials q and the image in the plane of the first row of Φ, parametrized by polar coordinates, which we call the Kepler transform. As an application among others, let C 1 ⊂ L 2 ( [ 0 , 2 π ] ) be the set of potentials q for which the equation − u ″ + q u = 0 admits a nonzero periodic solution: C 1 is diffeomorphic to the disjoint union of a hyperplane and Cartesian products of the usual cone in R 3 with H .

Flowlines transverse to knot and link fibrations

A knot or link in S 3 is fibred if the complement fibres over S 1 , the fibres being spanning surfaces. We focus on those fibred knots and links which have the following property: every vector field transverse to the fibres possesses closed flow lines of all possible knot and link types in S 3 . Our main result is that a large class of fibred knots and links has this property, including all fibred nontorus 2-bridge knots. In general, sufficient conditions include a pseudo-Anosov type monodromy map and a sufficiently high degree of symmetry.

Nonintegrability criteria for a class of differential equations with two regular singular points

Criteria for nonintegrability (in the sense of inexistence of continuous, single-valued first integrals in the complex domain) are given for first-order differential equations whose linear part is a homogeneous equation with several regular singular points (in a bounded domain of the complex plane). The method used is the poly-Painlevé test. Local equivalence maps between the nonlinear equations and their linear part are used in the proofs, and the (noncommutative) group of monodromy maps is studied to establish dense branching of solutions, hence nonintegrability. A dicussion concerning the notion of first integral is included.

A NOTE ON THURSTON-WINKELNKEMPER'S CONSTRUCTION OF CONTACT FORMS ON 3-MANIFOLDS

A contact structure on a closed oriented 3-manifold 3 is a completely nonintegrable plane field on it. The existence of a contact structure on 3 was first proved by Martinet [14] and then Lutz [12], [13] for each homotopy class of plane fields. A co-orientable contact structure is given as the kernel of a 1-form α with α ∧ α > 0 or < 0, which is called a positive or negative contact form respectively. Thurston and Winkelnkemper [17] deduced the existence of a contact form in an elegant way from Alexander’s theorem [1] on open-book decompositions. Eliashberg [2] showed that the most of these contact structures are, however, too flexible for geometrical interest (see §2). On the other hand, he completely characterized more historic examples related closely to the complex analysis in several variables, namely, the strictly pseudo-convex boundary of compact Stein surfaces ([4]). The symplectic fillability is one of its significant generalizations. Recently Loi and Piergallini [11] translated Eliashberg’s characterization of compact Stein surfaces into the language of Lefschetz fibration by using Gompf’s method [8]. They also showed that 3 is realizable as the boundary of a Stein surface if and only if it admits an open-book decomposition whose monodromy map is a composition of right-handed Dehn-twists. In this note, we improve Thurston-Winkelnkemper’s construction so that we obtain a symplectically fillable contact structure in the case where the monodromy map of a given open-book decomposition is a composition of right-handed Dehn-twists along mutually disjoint curves (Theorem 2). An interesting by-product of our construction is Theorem 3 in §4 which gives a deformation of symplectically fillable contact structures into a foliation with a Reeb component. Note that a foliation admitting a Reeb component itself is not symplectically fillable. I would like to thank the referee for noticing the importance of Theorem 3.

A Complete Description of Normal Surfaces for Infinite Series of 3-Manifolds

The set of all normal surfaces in a 3-manifold is a partial monoid under addition with a minimal generating set of fundamental surfaces. The available algorithm for finding the system of fundamental surfaces is of a theoretical nature and admits no implementation in practice. In this article, we give a complete and geometrically simple description for the structure of partial monoids for normal surfaces in lens spaces, generalized quaternion spaces, and Stallings manifolds with fiber a punctured torus and a hyperbolic monodromy map.

Stability of discrete breathers

We approach the problem of linear stability of discrete breathers in a rigorous way for networks of not necessarily identical oscillators with general type of coupling. In the case of Hamiltonian systems, using symplectic signature theory we give general conditions on the spectrum of the monodromy map in the uncoupled limit such that the discrete breathers are l 2-linearly stable for weak enough coupling. We consider also dissipative networks of oscillators. In this case we prove that the discrete breathers are not only stable but also attract a neighbourhood of initial data for any choice of l p -topology. Some examples are considered amongst which an instance of “roto-breather”.

Double Complexes and Euler L-Factors

In this paper we take into consideration a conjecture of Bloch equating in a suitable range and under some standard conjectures, the rank of the motivic cohomology of the special fiber of a semistable degeneration over the ring of integers of a number field with the order of zero of the local Euler factor of the L-function over the semistable prime corresponding to the special fiber. We then develop, following the philosophy that the fiber ‘at infinity’ of an arithmetic variety should be considered as ‘maximally degenerate’, a construction that goes in parallel to the one we use for the non-archimedean fiber. Namely, a definition of a double complex and a weighted operator N on it that plays the role of the logarithm of the (local) monodromy map at infinity. We like to see this archimedean theory as the analogue of the limiting mixed Hodge structure theory of a degeneration of projective varieties over a disc. In particular, this yields to a description of the (motivic) Deligne cohomology as homology of the mapping cone of N. The main result arising from this construction is the proof of a conjecture of Deninger. Namely we show that the archimedean Γ-factor of the Zeta-function of the archimedean fiber can be seen as the characteristic polynomial of an archimedean Frobenius acting on the subgroup of the invariants of N into the hypercohomology of our double complex.

Nonlocal quadratic Poisson algebras, monodromy map, and Bogoyavlensky lattices

A new Lax representation for the Bogoyavlensky lattice is found and its r-matrix interpretation is elaborated. The r-matrix structure turns out to be related to a highly nonlocal quadratic Poisson structure on a direct sum of associative algebras. The theory of such nonlocal structures is developed and the Poisson property of the monodromy map is worked out in the most general situation. Some problems concerning the duality of Lax representations are raised.

Pole assignment for linear periodic systems by memoryless output feedback

We consider linear periodic discrete-time systems. We are interested in the problem of placing the poles of the monodromy map by means of periodic output feedback of the same or multiple period. It is well known that, in general, the poles of time-invariant systems cannot be assigned by constant output feedback. This is in contrast with what can be obtained in the context of time-variant systems. The main contribution of this paper is that periodic output feedback suffices for pole placement of periodic systems. >

Assignment of nonlinear sampled‐data dynamics using generalized hold function control

This paper considers the problem of assigning the dynamics of a nonlinear analytic system using nonlinear generalized sampled‐data hold function (GSHF) control, in the neighborhood of an equilibrium point. On every sampling interval, the control input consists of a nonlinear time‐periodic function applied to the sampled value of the state vector. The nonlinear monodromy map is the state transition map from one sample time to the next. It is shown that this map is arbitrarily assignable by GSHF feedback if and only if the linear part of the system is controllable. Two approaches are proposed to construct a GSHF controller that performs the assignment. The first approach matches the coefficients of the Taylor series expansion of the monodromy map around the equilibrium. The second approach interpolates an optimal control law at several points in the vicinity of the equilibrium. These approaches are illustrated on an example.

Fuchsian moduli on a Riemann surface—its Poisson structure and Poincaré-Lefschetz duality

The moduli space of Fuchsian projective connections on a closed Riemann surface admits a Poisson structure. The moduli space of projective monodromy representations on the punctured Riemann surface also admits a Poisson structure which arises from the Poincare-Lefschetz duality for cohomology. We shall show that the former Poisson structure coincides with the pull-back of the latter by the projective monodromy map. This result explains intrinsically why a Hamiltonian structure arises in the monodromy preserving deformation

On monodromy map

LetΓbe a Fuchsian group acting on the upper half-planeUand having signature{p,n,0;v1,v2,…,vn};2p−2+∑j=1n(1−1vj)&gt;0.LetT(Γ)be the Teichmüller space ofΓ. Then there exists a vector bundleℬ(T(Γ))of rank3p−3+noverT(Γ)whose fibre over a pointt∈T(Γ)representingΓtis the space of bounded quratic differentialsB2(Γt)forΓt. LetHom(Γ,G)be the set of all homomorphisms fromΓinto the Mbius groupG.For a given(t,ϕ)∈ℬ(T(Γ))we get an equivalence class of projective structures and a conjugacy class of a homomorphismx∈Hom(Γ,G). Therefore there is a well defined mapΦ:ℬ(T(Γ))→Hom(Γ,G)/G,Φis called the monodromy map. We prove that the monromy map is hommorphism. The casen=0gives the previously known result by Earle, Hejhal Hubbard.

A uniqueness theorem of reflectable deformations of a Fuchsian group

Let Γ \Gamma be a Fuchsian group of signature ( p , n , m ; ν 1 , ν 2 , … , ν n ) (p,n,m;{\nu _1},{\nu _2}, \ldots ,{\nu _n}) ; 2 p − 2 + m + ∑ j = 1 n ( 1 − 1 / ν j ) &gt; 0 2p - 2 + m + \sum \nolimits _{j = 1}^n {(1 - 1/{\nu _j}) &gt; 0} . Let I 1 , I 2 , … , I m {I_1},{I_2}, \ldots ,{I_m} be a maximal set of inequivalent components of Ω ∩ R ^ \Omega \cap {\mathbf {\hat R}} ; Ω \Omega is the region of discontinuity and R ^ {\mathbf {\hat R}} is the extended real line. Let ϕ \phi be a quadratic differential for Γ \Gamma . Let f f be a solution of the Schwarzian differential equation S f = ϕ Sf = \phi . If ϕ \phi is reflectable, f f maps each I j {I_j} into a circle C j {C_j} . For each γ ∈ Γ \gamma \in \Gamma there is a Moebius transformation X ( γ ) \mathcal {X}(\gamma ) such that f ∘ γ = X ( γ ) ∘ f f \circ \gamma = \mathcal {X}(\gamma ) \circ f . We prove that ϕ \phi is determined by the homomorphism X \mathcal {X} and the circles C 1 , C 2 , … , C m {C_1},{C_2}, \ldots ,{C_m} .

On the Branch Points in the Branched Coverings of Links

AbstractLet l be a polygonal link in a 3-sphere S3 and a branched covering of l, which depends on the choice of a monodromy map ϕ. Let be the link in over l. In this paper we determine the exact position of in for some cases. For instance, if l is a torus link ((n + 1)p, n) and ϕ is an appropriate monodromy map of the fundamental group of S3 - l into the symmetric group of degree n + 1, then is an S3 and l is a torus link (np,n2). The 3-fold irregular branched covering of a doubled knot k is an S3, if it exists. The position of the link over k is shown in a figure. The link over knot 61 is obtained by K. A. Perko and the author, independently, and shown without proof in a paper by R. H. Fox [Can. J. Math. 22 (1970), 193-201]. The result mentioned in the above includes this case.

There Exist Inequivalent Knots With the Same Complement

The study and classification of knots has been based upon invariants of the knot complement. In this paper we give examples of inequivalent smooth spherical knots with diffeomorphic complements. These examples will also not be reflections, inversions or reflected inversions of each other. The knots we consider can be described as the class of all fibred knots with fibre a punctured n-torus, n > 3, and with the monodromy map on first homology having no negative eigenvalues. With easy modifications the arguments show that the constructed examples are not even piecewise linearly or topologically equivalent. A smooth n-knot is a smooth submanifold K of the (n + 2)-sphere S+2, homeomorphic to So, and if K is diffeomorphic to S, it is said that the

The monodromy of a curve with ordinary double points

Mutsuo Oka (Bures-sur-Yvette) w 1. Introduction Let f(x,y,z) be a square-free homogeneous polynomial of degree d and let C be the projective curve in IP 2 which is defined by C=f-1(0). We want to study gt(lP 2 - C). For this we consider the Milnor fibering of f: f/[fl=arg(f): Ss-K~S 1 where K=f-~(O)c~S 5. The fiber F of this fibering is naturally diffeomorphic to any affine hypersurfaee X~=f-l(t)ctl? 3 (t+-O). Let h: F~F be the monodromy map which is defined by