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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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A framework toward understanding the characterization of holomorphic dynamics

A framework toward understanding the characterization of holomorphic dynamics

Chapter:
(p.235) A framework toward understanding the characterization of holomorphic dynamics
Source:
Frontiers in Complex Dynamics
Author(s):

Yunping Jiang

, Araceli Bonifant, Mikhail Lyubich, Scott Sutherland
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691159294.003.0012

This chapter reviews the characterization of geometrically finite rational maps and then outlines a framework for characterizing holomorphic maps. Whereas Thurston's methods are based on estimates of hyperbolic distortion in hyperbolic geometry, the framework suggested here is based on controlling conformal distortion in spherical geometry. The new framework enables one to relax two of Thurston's assumptions: first, that the iterated map has finite degree and, second, that its post-critical set is finite. Thus, it makes possible to characterize certain rational maps for which the post-critical set is not finite as well as certain classes of entire and meromorphic coverings for which the iterated map has infinite degree.

Keywords:   holomorphic dynamics, geometrically finite rational maps, hyperbolic distortion, hyperbolic geometry, conformal distortion, spherical geometry

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