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Commuting rational functions revisited

Part of: Complex dynamical systems Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  15 August 2019

FEDOR PAKOVICH Open the ORCID record for FEDOR PAKOVICH [Opens in a new window]
FEDOR PAKOVICH*
Affiliation:
Department of Mathematics, Ben Gurion University, P.O. Box 653, Beer Sheva, 8410501, Israel email pakovich@math.bgu.ac.il
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Abstract

Let $B$ be a rational function of degree at least two that is neither a Lattès map nor conjugate to $z^{\pm n}$ or $\pm T_{n}$. We provide a method for describing the set $C_{B}$ consisting of all rational functions commuting with $B$. Specifically, we define an equivalence relation $\underset{B}{{\sim}}$ on $C_{B}$ such that the quotient $C_{B}/\underset{B}{{\sim}}$ possesses the structure of a finite group $G_{B}$, and describe generators of $G_{B}$ in terms of the fundamental group of a special graph associated with $B$.

Keywords

commuting rational functionsthe Ritt theoremcommon iterates

MSC classification

Primary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions
Secondary: 37F10: Polynomials; rational maps; entire and meromorphic functions

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 1 , January 2021 , pp. 295 - 320
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Copyright
© Cambridge University Press, 2019

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References

Eremenko, A.. Some functional equations connected with the iteration of rational functions. Leningrad Math. J. 1 (1990), 905–919.Google Scholar
Fatou, P.. Sur l’iteration analytique et les substitutions permutables. J. Math. Pures Appl. (9) 2(1923) 343–384.Google Scholar
Julia, G.. Mémoire sur la permutabilité des fractions rationelles. Ann. Sci. Éc. Norm. Supér (4) 39(3) (1922), 131–215.10.24033/asens.740Google Scholar
Muzychuk, M. and Pakovich, F.. Jordan-Holder theorem for imprimitivity systems and maximal decompositions of rational functions. Proc. Lond. Math. Soc. (3) 102(1) (2011), 1–24.10.1112/plms/pdq009Google Scholar
Pakovich, F.. Finiteness theorems for commuting and semiconjugate rational functions. Preprint, 2015, arXiv:1604:04771.Google Scholar
Pakovich, F.. On semiconjugate rational functions. Geom. Funct. Anal. 26 (2016), 1217–1243.10.1007/s00039-016-0383-6Google Scholar
Pakovich, F.. Polynomial semiconjugacies, decompositions of iterations, and invariant curves. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) XVII (2017), 1417–1446.Google Scholar
Pakovich, F.. Semiconjugate rational functions: a dynamical approach. Arnold Math. J. 4(1) (2018), 59–68.10.1007/s40598-018-0081-6Google Scholar
Pakovich, F.. Recomposing rational functions. Int. Math. Res. Not. IMRN 2019(7) (2019), 1921–1935.10.1093/imrn/rnx172Google Scholar
Pakovich, F.. On generalized Latès maps. J. Anal. Math. accepted.Google Scholar
Ritt, J.. Prime and composite polynomials. Amer. Math. Soc. Transl. Ser. 2 23 (1922), 51–66.10.1090/S0002-9947-1922-1501189-9Google Scholar
Ritt, J. F.. On the iteration of rational functions. Trans. Amer. Math. Soc. 21(3) (1920), 348–356.10.1090/S0002-9947-1920-1501149-6Google Scholar
Ritt, J. F.. Permutable rational functions. Trans. Amer. Math. Soc. 25 (1923), 399–448.10.1090/S0002-9947-1923-1501252-3Google Scholar
Stillwell, J.. Classical Topology and Combinatorial Group Theory (Graduate Texts in Mathematics, 72). Springer, New York, 1993.Google Scholar