Skip to main content
Log in

On Mutually Semiconjugate Rational Functions

  • Research Contribution
  • Published:
Arnold Mathematical Journal Aims and scope Submit manuscript

Abstract

We characterize pairs of rational functions A, B such that A is semiconjugate to B, and B is semiconjugate to A.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Buff, X., Epstein, A.: From local to global analytic conjugacies. Ergodic Theory Dyn. Syst. 27(4), 1073–1094 (2007)

    Article  MathSciNet  Google Scholar 

  • Douady, A., Hubbard, J.: A proof of Thurston’s topological characterization of rational functions. Acta Math. 171(2), 263–297 (1993)

    Article  MathSciNet  Google Scholar 

  • Eremenko, A.: Some functional equations connected with the iteration of rational functions (Russian). Algebra I Anal. 1, 102–116 (1989). (translation in Leningrad Math. J. 1(1990), 905–919)

    Google Scholar 

  • Eremenko, A.: Invariant curves and semiconjugacies of rational functions. Fund. Math. 219(3), 263–270 (2012)

    Article  MathSciNet  Google Scholar 

  • Fatou, P.: Sur l’iteration analytique et les substitutions permutables. J. Math. Pures Appl. 2, 343–384 (1923)

    MATH  Google Scholar 

  • Julia, G.: Mémoire sur la permutabilité des fractions rationelles. Ann. Sci. École Norm. Sup. 39(3), 131–215 (1922)

    Article  MathSciNet  Google Scholar 

  • McMullen, C.: Families of rational maps and iterative root-finding algorithms. Ann. Math. 125(3), 467–493 (1987)

    Article  MathSciNet  Google Scholar 

  • Medvedev, A., Scanlon, T.: Invariant varieties for polynomial dynamical systems. Ann. Math. 179(1), 81–177 (2014)

    Article  MathSciNet  Google Scholar 

  • Milnor, J.: On Lattès maps, dynamics on the Riemann sphere. Eds. P. Hjorth and C. L. Petersen. A Bodil Branner Festschrift. Eur. Math. Soc. 2006, 9–43 (2006)

    Google Scholar 

  • Milnor, J.: Dynamics in One Complex Variable, Princeton Annals in Mathematics 160. Princeton University Press, Princeton (2006)

    Google Scholar 

  • Pakovich, F.: On semiconjugate rational functions. Geom. Funct. Anal 26, 1217–1243 (2016)

    Article  MathSciNet  Google Scholar 

  • Pakovich, F.: Polynomial semiconjugacies, decompositions of iterations, and invariant curves. Ann. Sci. Norm. Super. Pisa Cl. Sci. 5(XVII), 1417–1446 (2017)

    MathSciNet  MATH  Google Scholar 

  • Pakovich, F.: Semiconjugate rational functions: a dynamical approach. Arnold Math. J. 4(1), 59–68 (2018)

    Article  MathSciNet  Google Scholar 

  • Pakovich, F.: Recomposing rational functions. Int. Math. Res. Not. 7, 1921–1935 (2019a)

    Article  MathSciNet  Google Scholar 

  • Pakovich, F.: On generalized Lattès maps. J. Anal. Math. (2019b)

  • Pakovich, F.: Commuting rational functions revisited. Ergodic Theory Dyn. Syst. (2019c). https://doi.org/10.1017/etds.2019.51

  • Pakovich, F.: Finiteness theorems for commuting and semiconjugate rational functions, preprint. arxiv:1604.04771 (2019d)

  • Ritt, J.F.: Permutable rational functions. Trans. Am. Math. Soc. 25, 399–448 (1923)

    Article  MathSciNet  Google Scholar 

  • Silverman, J.: The Arithmetic of Dynamical Systems, Graduate Texts in Mathematics, p. 241. Springer, New York (2007)

    Book  Google Scholar 

Download references

Acknowledgements

This research was supported by the ISF Grant no. 1432/18.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to F. Pakovich.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

To Rafail Kalmanovich Gordin, on the occasion of his 70th birthday.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pakovich, F. On Mutually Semiconjugate Rational Functions. Arnold Math J. 5, 339–354 (2019). https://doi.org/10.1007/s40598-019-00124-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40598-019-00124-9

Keywords

Navigation