Monodromy Groups and Linearly Polymorphic Functions

Hejhal, Dennis A.

doi:10.1515/9781400881642-020

Cited by 27 publications

(40 citation statements)

References 12 publications

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“…Mess introduces these notions briefly, and shows how the Bers embedding gives a holomorphic section of the bundle P (S) → Teich(S). His assertion that the holonomy map from P (S) to the P SL(2, C) representation variety is a holomorphic submersion is due to Hejhal [64].…”

Section: N6 the Case Of De Sitter Spacementioning

confidence: 99%

Notes on a paper of Mess

et al. 2007

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Section: N6 the Case Of De Sitter Spacementioning

confidence: 99%

Notes on a paper of Mess

et al. 2007

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“…In general, the explicit calculation of the eigenvalues of the monodromy matrices is an unsuccessful business except rare cases such as Hamiltonian systems with homogeneous polynomial functions [17], Riemann's equation, and the Jordan-Pochhammer equations [10,4,2]. This is one of the unavoidable difficulties in performing Ziglin's analysis of general dynamical systems.…”

Section: Singularity Analysis Of Nonlinear Latticesmentioning

confidence: 99%

Non-perturbative non-integrability of non-homogeneous nonlinear lattices induced by non-resonance hypothesis

Umeno

1996

Physica D: Nonlinear Phenomena

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We have succeeded in applying Ziglin's test on non-integrability to nonhomogeneous nonlinear lattices (Fermi-Pasta-Ulam lattices). By explicit calculations of the eigenvalues of the monodromy matrices concerning the normal variational equations of Lamé type and with the use of the non-resonance hypothesis about the eigenvalues, we obtained a theorem proving the nonexistence of additional analytic conserved quantities other than the Hamiltonian itself for FPU lattices in the low energy limit. Furthermore, after introducing a concept of degree of non-integrability, we have investigated the classification of non-homogeneous nonlinear lattices using a transformation ℜ from a non-homogeneous nonlinear lattice to another non-homogeneous nonlinear lattice, which preserves their degree of non-integrability.

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“…230, §50], with a modern history developed by many authors ( [Ma69], [He75], [Fa83], [ST83], [Go87], [GKM00], [Ta97], [Mc98]). The main technical tool in our proof that bending measures give coordinates for Bers slices, and the second major goal of this paper, is the completion of the proof of the "Grafting Conjecture".…”

Section: §1 Introductionmentioning

confidence: 99%

“…This identifies P g with the total space of the bundle Q g → T g of holomorphic quadratic differentials over Teichmüller space. This identification is representative of the complex analytic side of the theory; see for instance [Ea81], [Gu81], [He75], [Kr69], [Kr71], [KM81], [MV94], [Sh87], [ST99].…”

Section: §1 Introductionmentioning

confidence: 99%

The grafting map of Teichmüller space

Scannell

Wolf

2002

J. Amer. Math. Soc.

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IntroductionOne of the underlying principles in the study of Kleinian groups is that aspects of the complex projective geometry of quotients ofĈ by these groups reflect properties of the three-dimensional hyperbolic geometry of the quotients of H 3 by these groups. Yet, even though it has been over thirty-five years since Lipman Bers exhibited a holomorphic embedding of the Teichmüller space of Riemann surfaces in terms of the projective geometry of a Teichmüller space of quasi-Fuchsian manifolds, no corresponding embedding in terms of the three-dimensional hyperbolic geometry has been presented. One of the goals of this paper is to give such an embedding. This embedding is straightforward and has been expected for some time ([Ta97], [Mc98]): to each member of a Bers slice of the space QF of quasi-Fuchsian 3-manifolds, we associate the bending measured lamination of the convex hull facing the fixed "conformal" end.The geometric relationship between a boundary component of a convex hull and the projective surface at infinity for its end is given by a process known as grafting, an operation on projective structures on surfaces that traces its roots back at least to Klein [Kl33, p. 230 ). The main technical tool in our proof that bending measures give coordinates for Bers slices, and the second major goal of this paper, is the completion of the proof of the "Grafting Conjecture". This conjecture states that for a fixed measured lamination λ, the self-map of Teichmüller space induced by grafting a surface along λ is a homeomorphism of Teichmüller space; our contribution to this argument is a proof of the injectivity of the grafting map. While the principal application of this result that we give is to geometric coordinates on the Bers slice of QF , one expects that the grafting homeomorphism might lead to other systems of geometric coordinates for other families of Kleinian groups (see §5.2); thus we feel that this result is of interest in its own right.We now state our results and methods more precisely. Throughout, S will denote a fixed differentiable surface which is closed, orientable, and of genus g ≥ 2. Let T g be the corresponding Teichmüller space of marked conformal structures on S, and let P g denote the deformation space of (complex) projective structures on S (see §2 for definitions).

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Monodromy Groups and Linearly Polymorphic Functions

Cited by 27 publications

References 12 publications

Notes on a paper of Mess

Notes on a paper of Mess

Non-perturbative non-integrability of non-homogeneous nonlinear lattices induced by non-resonance hypothesis

The grafting map of Teichmüller space

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