Arnold Mathematical Journal

, Volume 4, Issue 1, pp 59–68 | Cite as

Semiconjugate Rational Functions: A Dynamical Approach

  • F. Pakovich
Research Contribution


Using dynamical methods we give a new proof of the theorem saying that if ABX are rational functions of complex variable z of degree at least two such that \(A\circ X=X\circ B\) and \({\mathbb C}(B,X)={\mathbb C}(z)\), then the Galois closure of the field extension \({\mathbb C}(z)/{\mathbb C}(X)\) has genus zero or one.


Semiconjugate rational functions Poincaré functions Invariant curves Galois closure Orbifolds 



The author is grateful to A. Eremenko for discussions.


  1. Beardon, A., Ng, T.W.: Parametrizations of algebraic curves. Ann. Acad. Sci. Fenn. Math. 31(2), 541–554 (2006)MathSciNetMATHGoogle Scholar
  2. Buff, X., Epstein, A.: From local to global analytic conjugacies. Ergod. Theory Dyn. Syst. 27(4), 1073–1094 (2007)MathSciNetCrossRefMATHGoogle Scholar
  3. Eremenko, A.: Invariant curves and semiconjugacies of rational functions. Fundamenta Math. 219(3), 263–270 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. Eremenko, A.: Some functional equations connected with the iteration of rational functions (Russian). Algebra i Analiz 1, 102–116 (1989). (translation in Leningrad Math. J. 1 (1990), 905–919)Google Scholar
  5. Fried, M.: Introduction to modular towers: generalizing dihedral group-modular curve connections. Recent developments in the inverse Galois problem, 111–171, Contemp. Math., 186, Am. Math. Soc., Providence, RI (1995)Google Scholar
  6. Fatou, P.: Sur l’iteration analytique et les substitutions permutables. J. Math. Pures Appl. 9(2), 343–384 (1923)MATHGoogle Scholar
  7. Julia, G.: Mémoire sur la permutabilité des fractions rationelles. Ann. Sci. École Norm. Sup. 39(3), 131–215 (1922)MathSciNetCrossRefMATHGoogle Scholar
  8. Medvedev, A., Scanlon, T.: Invariant varieties for polynomial dynamical systems. Ann. Math. 179(1), 81–177 (2014)MathSciNetCrossRefMATHGoogle Scholar
  9. Milnor, J.: Dynamics in One Complex Variable, vol. 160. Princeton Annals in Mathematics. Princeton University Press, Princeton (2006)Google Scholar
  10. Pakovich, F.: On semiconjugate rational functions. Geom. Funct. Anal. 26, 1217–1243 (2016a)Google Scholar
  11. Pakovich, F.: Finiteness theorems for commuting and semiconjugate rational functions. (2016b). arXiv:1604:04771
  12. Pakovich, F.: Polynomial semiconjugacies, decompositions of iterations, and invariant curves. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) XVII, 1417–1446 (2017a)MathSciNetMATHGoogle Scholar
  13. Pakovich, F.: Recomposing rational functions. Internat. Math. Res. Notices (2017b).
  14. Pakovich, F.: On rational functions whose normalization has genus zero or one. Acta Arith. 182, 73–100 (2018a)MathSciNetCrossRefMATHGoogle Scholar
  15. Pakovich, F.: On generalized Lattès maps. (2018b). arXiv:1612.01315v3 (preprint)
  16. Picard, E.: Démonstration d’un thèoréme général sur les fonctions uniformes liées par une relation algèbrique. Acta Math. 11(1–4), 1–12 (1887)MathSciNetCrossRefMATHGoogle Scholar
  17. Ritt, J.F.: Prime and composite polynomials. Trans. Am. Math. Soc. 23, 51–66 (1922)MathSciNetCrossRefMATHGoogle Scholar
  18. Ritt, J.F.: Permutable rational functions. Trans. Am. Math. Soc. 25, 399–448 (1923)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2018

Authors and Affiliations

  1. 1.Department of MathematicsBen Gurion University of the NegevBeer ShebaIsrael

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