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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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Entropy for hyperbolic Riemann surface laminations I

Entropy for hyperbolic Riemann surface laminations I

(p.569) Entropy for hyperbolic Riemann surface laminations I
Frontiers in Complex Dynamics

Tien-Cuong Dinh

Viet-Anh Nguyen

Nessim Sibony

, Araceli Bonifant, Mikhail Lyubich, Scott Sutherland
Princeton University Press

This chapter introduces a notion of entropy for possibly singular hyperbolic laminations by Riemann surfaces. It also studies the transverse regularity of the Poincaré metric and the finiteness of the entropy. The chapter first focuses on compact laminations, which are transversally smooth, before turning to the case of singular foliations, showing how the Poincaré metric on leaves is transversally Hölder continuous. In addition, the chapter considers the problem in the proof that the entropy is finite for singular foliations is quite delicate and requires a careful analysis of the dynamics around the singularities. Finally, the chapter discusses a notion of metric entropy for harmonic probability measures and gives some open questions.

Keywords:   entropy, Riemann surfaces, hyperbolic laminations, Poincaré metric, finiteness, compact laminations, singular foliations, metric entropy, harmonic probability measures

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