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Frontiers in Complex DynamicsIn Celebration of John Milnor's 80th Birthday$
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Araceli Bonifant, Misha Lyubich, and Scott Sutherland

Print publication date: 2014

Print ISBN-13: 9780691159294

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691159294.001.0001

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Leading monomials of escape regions

Leading monomials of escape regions

Chapter:
(p.103) Leading monomials of escape regions
Source:
Frontiers in Complex Dynamics
Author(s):

Jan Kiwi

, Araceli Bonifant, Mikhail Lyubich, Scott Sutherland
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691159294.003.0007

This chapter considers a leading monomial vector, which uniquely determines the escape region and within which is encoded important information about the limiting behavior of the periodic critical orbit. Thus, for each p ≥ 1, this chapter considers the affine algebraic curve Sₚ formed by all monic and centered cubic polynomials with a marked critical point which has period p under iterations. Each unbounded hyperbolic component of Sₚ, called an escape region, has an associated vector of leading monomials which encodes the asymptotic behavior of the periodic critical orbit. The chapter shows that this vector determines the escape region, giving a positive answer to a question posed by Bonifant and Milnor.

Keywords:   leading monomials, leading monomial vector, escape regions, asymptotic behavior, periodic critical orbit, unbounded hyperbolic components

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