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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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Milnor's conjecture on monotonicity of topological entropy: Results and questions

Milnor's conjecture on monotonicity of topological entropy: Results and questions

Chapter:
(p.323) Milnor's conjecture on monotonicity of topological entropy: Results and questions
Source:
Frontiers in Complex Dynamics
Author(s):

Sebastian van Strien

, Araceli Bonifant, Mikhail Lyubich, Scott Sutherland
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691159294.003.0014

This chapter discusses Milnor's conjecture on monotonicity of entropy and gives a short exposition of the ideas used in its proof. It discusses the history of this conjecture, gives an outline of the proof in the general case, and describes the state of the art in the subject. The proof makes use of an important result by Kozlovski, Shen, and van Strien on the density of hyperbolicity in the space of real polynomial maps, which is a far-reaching generalization of the Thurston Rigidity Theorem. (In the quadratic case, density of hyperbolicity had been proved in studies done by M. Lyubich and J. Graczyk and G. Swiatek.) The chapter concludes with a list of open problems.

Keywords:   Milnor's conjecture, monotonicity, topological entropy, hyperbolicity, Thurston Rigidity Theorem, entropy

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