Computable Riemann Surfaces

(Extended Abstract)
  • Robert Rettinger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4497)


In this paper we introduce computable and time bounded Riemann surfaces, based on the classical abstract definition by charts. Building upon this definition we discuss computable versions of several classical results, such as the existence of complete continuations of holomorphic functions, universal coverings and the uniformization theorem (for some cases).

Though we state most of our results for computable surfaces, many of them can also be transformed to a uniform version, i.e. based on representations of the class of Riemann surfaces (modulo conformal equivalence).


computable Riemann surface computable uniformization 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ahl81.
    Ahlfors, L.: Complex Analysis. McGraw-Hill, New York (1981)Google Scholar
  2. BBY06.
    Binder, I., Braverman, M., Yampolsky, M.: On computational complexity of Riemann mapping (unpublished manuscript)Google Scholar
  3. BBY07.
    Binder, I., Braverman, M., Yampolsky, M.: On computational complexity of Siegel Julia sets, Commun. Math. PhysicsGoogle Scholar
  4. Bra06.
    Braverman, M.: Parabolic Julia Sets are Polynomial Time Computable, Nonlinearity 19 (2006)Google Scholar
  5. BY06.
    Braverman, M., Yampolsky, M.: Non-Computable Julia Sets. Journ. Amer. Math. Soc. 19(3) (2006)Google Scholar
  6. Her99.
    Hertling, P.: An effective Riemann mapping theorem. Theor. Comp. Sci. 219, 225–265 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Hoe06.
    van der Hoeven, J.: On Effective Analytic Continuation 2006 (unpublished manuscript)Google Scholar
  8. Mue87.
    Müller, N.: Uniform computational complexity of Taylor series. In: Ottmann, T. (ed.) Automata, Languages and Programming. LNCS, vol. 267, pp. 435–444. Springer, Berlin (1987)CrossRefGoogle Scholar
  9. Mue93.
    Müller, N.: Polynomial-Time Computation of Taylor Series In: Proc. 22 JAIIO - PANEL ’93, Part 2, Buenos Aires, pp. 259–281 (1993)Google Scholar
  10. Rettinger, R., Weihrauch, K.: The computational complexity of some julia sets, STOC, pp. 177–185 (2003)Google Scholar
  11. Ret05.
    Rettinger, R.: A Fast Algorithm for Julia Sets of Hyperbolic Rational Functions. ENTCS 120, 145–157 (2005)MathSciNetGoogle Scholar
  12. Ret07.
    Rettinger, R.: On continuations of holomorphic mappings, 2007 (unpublished manuscript)Google Scholar
  13. Wei00.
    Weihrauch, K.: Computable Analysis. Springer, Berlin (2000)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Robert Rettinger
    • 1
  1. 1.FernUniversität Hagen, LG Komplexität und Algorithmen, Universitätsstrasse 1, D-58095 HagenGermany

Personalised recommendations