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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 II

Entropy for hyperbolic Riemann surface laminations II

Chapter:
(p.593) Entropy for hyperbolic Riemann surface laminations II
Source:
Frontiers in Complex Dynamics
Author(s):

Tien-Cuong Dinh

Viet-Anh Nguyen

Nessim Sibony

, Araceli Bonifant, Mikhail Lyubich, Scott Sutherland
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691159294.003.0021

This chapter studies Riemann surface foliations with tame singular points. It shows that the hyperbolic entropy of a Brody hyperbolic foliation by Riemann surfaces with linearizable isolated singularities on a compact complex surface is finite. The chapter then proves the finiteness of the entropy in the local setting near a singular point in any dimension, using a division of a neighborhood of a singular point into adapted cells. Next, the chapter estimates the modulus of continuity for the Poincaré metric along the leaves of the foliation, using notion of conformally (R,δ‎)-close maps. The estimate holds for foliations on manifolds of higher dimension.

Keywords:   entropy, Riemann surfaces, Riemann surface foliations, tame singular points, hyperbolic entropy, Brody hyperbolic foliation, compact complex surface, finiteness, Poincaré metric

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