# 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* = 2*n* − 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 2*n* − 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.

Contents Full-Text PDF