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SPECIAL CURVES AND POSTCRITICALLY FINITE POLYNOMIALS

Part of:Geometric function theoryArithmetic algebraic geometryComplex dynamical systems

Published online by Cambridge University Press:  19 September 2013

MATTHEW BAKER and
LAURA DE MARCO
MATTHEW BAKER
Affiliation:
Georgia Institute of Technology, Mathematics Atlanta, GA, United Statesmbaker@math.gatech.edu
LAURA DE MARCO
Affiliation:
University of Illinois at Chicago, Mathematics Chicago, IL, United Statesdemarco@uic.edu

Abstract

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We study the postcritically finite maps within the moduli space of complex polynomial dynamical systems. We characterize rational curves in the moduli space containing an infinite number of postcritically finite maps, in terms of critical orbit relations, in two settings: (1) rational curves that are polynomially parameterized; and (2) cubic polynomials defined by a given fixed point multiplier. We offer a conjecture on the general form of algebraic subvarieties in the moduli space of rational maps on ${ \mathbb{P} }^{1} $ containing a Zariski-dense subset of postcritically finite maps.

MSC classification

Secondary:37F45: Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations11G50: Heights30C10: Polynomials

Information

Type
Research Article
Information
Forum of Mathematics, Pi , Volume 1 , 2013, e3
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
Copyright
© The Author(s) 2013.

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