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March 25, 2015 By

Parameterizing Degree $n$ Polynomials by Multipliers of Periodic Orbits

C. R. Math. Rep. Acad. Sci. Canada Vol. 35 (4) 2013, pp. 148–154

December 31, 2013

Igors Gorbovickis , Department of Mathematics, University of Toronto, Room 6290, 40 St. George Street, Toronto, Ontario, Canada M5S 2E4; e-mail: igors.gorbovickis@utoronto.ca

Abstract/Résumé:

We present the following result: consider the space of complex polynomials of degree $n\ge 3$ with $n-1$ distinct marked periodic orbits of given periods. Then this space is irreducible and the multipliers of the marked 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 $n$ polynomial maps can be locally parameterized by the multipliers of $n-1$ arbitrary distinct periodic orbits. A detailed proof of this result (together with a proof of a more general statement) is given in [2]. In this exposition we substitute some of the technical lemmas from [2] with more geometric arguments.

On démontre le résultat suivant: considérons l’espace des polynômes complexes de degré $n\ge 3$ avec $n-1$ orbites périodiques distinctes marquées des périodes données. Alors cet espace est irréductible et considéré comme fonctions algébriques sur l’espace mentionné, les multiplicateurs des orbites périodiques marquées sont algébriquement indépendant sur $\mathbb{C}$. Ceci est équivalent á dire que á son point générique, l’espace module des applications polynomiales de degré $n$ peut être localement paramétrer par les multiplicateurs de $n-1$ orbites périodiques distinctes quelconques. Une démonstration de ce résultat (avec une preuve d’un énoncé plus général) est donnée en [2]. Dans cette exposition on remplace quelques lemmes techniques avec des arguments plus géométriques.

Keywords:
AMS Subject Classification: Polynomials; rational maps; entire and meromorphic functions 37F10

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