Moscow Mathematical Journal

Volume 15, Issue 1, January–March 2015 pp. 73–87.

Algebraic Independence of Multipliers of Periodic Orbits in the Space of Rational Maps of the Riemann Sphere

Authors: Igors Gorbovickis
Author institution:Department of Mathematics, University of Toronto, Room 6290, 40 St. George Street, Toronto, Ontario, Canada M5S 2E4

Summary:

We consider the space of degree n ≥ 2 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 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.

2010 Math. Subj. Class. 37F10, 37F05.

Keywords: Rational maps of the Riemann sphere, multipliers of periodic orbits.

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