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 Mosc. Math. J., 2015, Volume 15, Number 1, Pages 73–87 (Mi mmj549)

Algebraic independence of multipliers of periodic orbits in the space of rational maps of the Riemann sphere

Igors Gorbovickis

Department of Mathematics, University of Toronto, Room 6290, 40 St. George Street, Toronto, Ontario, Canada M5S 2E4

Abstract: We consider the space of degree $n\ge2$ rational maps of the Riemann sphere with $k$ distinct marked periodic orbits of given periods. First, we show that this space is irreducible. For $k=2n-2$ and with some mild restrictions on the periods of the marked periodic orbits, we show that the multipliers of these periodic orbits, considered as algebraic functions on the above mentioned space, are algebraically independent over $\mathbb C$. Equivalently, this means that at its generic point, the moduli space of degree $n$ rational maps can be locally parameterized by the multipliers of any $2n-2$ distinct periodic orbits, satisfying the above mentioned conditions on their periods. This work extends previous similar result obtained by the author for the case of complex polynomial maps.

Key words and phrases: rational maps of the Riemann sphere, multipliers of periodic orbits.

DOI: https://doi.org/10.17323/1609-4514-2015-15-1-73-87

Full text: http://www.mathjournals.org/.../2015-015-001-004.html
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MSC: 37F10, 37F05
Received: March 2, 2014; in revised form July 28, 2014
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Citation: Igors Gorbovickis, “Algebraic independence of multipliers of periodic orbits in the space of rational maps of the Riemann sphere”, Mosc. Math. J., 15:1 (2015), 73–87

Citation in format AMSBIB
\Bibitem{Gor15} \by Igors~Gorbovickis \paper Algebraic independence of multipliers of periodic orbits in the space of rational maps of the Riemann sphere \jour Mosc. Math.~J. \yr 2015 \vol 15 \issue 1 \pages 73--87 \mathnet{http://mi.mathnet.ru/mmj549} \crossref{https://doi.org/10.17323/1609-4514-2015-15-1-73-87} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3427412} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000354886200004} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. I. Gorbovickis, “Algebraic independence of multipliers of periodic orbits in the space of polynomial maps of one variable”, Ergodic Theory Dynam. Systems, 36:4 (2016), 1156–1166