Journal of Dynamical and Control Systems

, Volume 6, Issue 2, pp 219–264

On the Riemann–Hilbert Problem in Dimension 4

  • A. I. Gladyshev
Article

Abstract

We consider the problem of existence of a Fuchsian system with a prescribed 4-dimensional monodromy. We give a classification of all cases of negative solution of this problem in terms of reducibility pattern of the representation, its local structure (which is described by a modification of Jordan form), and restrictions on asymptotics of solutions to Fuchsian systems in lower dimensions. We also show that realization of a reducible 4-dimensional representation by a Fucshian system, if it exists, can be chosen in a block upper-triangular form (though not necessarily with the same reducibility pattern). At the end of the paper, we present new counterexamples to the Riemann–Hilbert problem in dimension 4.

Riemann–Hilbert problem Fuchsian system monodromy representation 

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References

  1. 1.
    A. A. Bolibruch, The Riemann–Hilbert problem. [Russian] Russ. Math. Surv. 45(1990), No. 2(272).Google Scholar
  2. 2.
    D.V. Anosov and A.A. Bolibruch, The Riemann-Hilbert problem. Aspects Math. Vol. E 22, Braunschweig, Wiesbaden: Vieweg, 1994.Google Scholar
  3. 3.
    A. H. M. Levelt, Hypergeometric functions. Nederl. Akad. Wetensch. Proc., Ser. A 64(1961).Google Scholar
  4. 4.
    V.P. Kostov, Quantum states of monodromy groups. J. Dynam. Control Syst. 5(1999), No. 1, 51-100.Google Scholar
  5. 5.
    ______, Fuchsian linear systems on ℂℙ1and the Riemann–Hilbert problem. C.R. Acad. Sci. Paris, Ser. 1 (1992), 143-148.Google Scholar
  6. 6.
    Vandamme Juliet, Simplification du thèoreme de Moser. In: Problème de Riemann–Hilbert pour une monodromie rèsoluble de dimension4, Universitède NICE, Prèpublication, No. 482, Mars 1997.Google Scholar
  7. 7.
    A. A. Bolibruch, On the problem of existence of Fuchsian systems with given asymptotics. Proc. Steklov Inst. Math. 216(1996), 32-44.Google Scholar
  8. 8.
    A. I. Gladyshev, On reducible Fuchsian systems of order 4. [Russian] In: Problems of Mathematics in Physical and Engeneering problems–a collection of articles, Moscow Institute of Physics and Technology, 1994.Google Scholar

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • A. I. Gladyshev
    • 1
  1. 1.Yukos Oil CompanyMoscowRussia

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