Conformal and Harmonic Measures on Laminations Associated with Rational Maps (Memoirs of the American Mathematical Society Series, No. 820) by Vadim A. Kaimanovich, Mikhail Lyubich | | 9780821836156 | Paperback | Barnes & Noble
Conformal and Harmonic Measures on Laminations Associated with Rational Maps (Memoirs of the American Mathematical Society Series, No. 820)

Conformal and Harmonic Measures on Laminations Associated with Rational Maps (Memoirs of the American Mathematical Society Series, No. 820)

by Vadim A. Kaimanovich, Mikhail Lyubich
     
 

ISBN-10: 0821836153

ISBN-13: 9780821836156

Pub. Date: 01/03/2005

Publisher: American Mathematical Society

This book is dedicated to Dennis Sullivan on the occasion of his 60th birthday. The framework of affine and hyperbolic laminations provides a unifying foundation for many aspects of conformal dynamics and hyperbolic geometry. The central objects of this approach are an affine Riemann surface lamination $\mathcal A$ and the associated hyperbolic 3-lamination

Overview

This book is dedicated to Dennis Sullivan on the occasion of his 60th birthday. The framework of affine and hyperbolic laminations provides a unifying foundation for many aspects of conformal dynamics and hyperbolic geometry. The central objects of this approach are an affine Riemann surface lamination $\mathcal A$ and the associated hyperbolic 3-lamination $\mathcal H$ endowed with an action of a discrete group of isomorphisms. This action is properly discontinuous on $\mathcal H$, which allows one to pass to the quotient hyperbolic lamination $\mathcal M$. Our work explores natural ''geometric'' measures on these laminations. We begin with a brief self-contained introduction to the measure theory on laminations by discussing the relationship between leafwise, transverse and global measures. The central themes of our study are: leafwise and transverse ''conformal streams'' on an affine lamination $\mathcal A$ (analogues of the Patterson-Sullivan conformal measures for Kleinian groups), harmonic and invariant measures on the corresponding hyperbolic lamination $\mathcal H$, the ''Anosov—Sinai cocycle'', the corresponding ''basic cohomology class'' on $\mathcal A$ (which provides an obstruction to flatness), and the Busemann cocycle on $\mathcal H$. A number of related geometric objects on laminations — in particular, the backward and forward Poincare series and the associated critical exponents, the curvature forms and the Euler class, currents and transverse invariant measures, $\lambda$-harmonic functions and the leafwise Brownian motion — are discussed along the lines. The main examples are provided by the laminations arising from the Kleinian and the rational dynamics. In the former case, $\mathcal M$ is a sublamination of the unit tangent bundle of a hyperbolic 3-manifold, its transversals can be identified with the limit set of the Kleinian group, and we show how the classical theory of Patterson-Sullivan measures can be recast in terms of our general approach. In the latter case, the laminations were recently constructed by Lyubich and Minsky in [LM97]. Assuming that they are locally compact, we construct a transverse $\delta$-conformal stream on $\mathcal A$ and the corresponding $\lambda$-harmonic measure on $\mathcal M$, where $\lambda=\delta(\delta-2)$. We prove that the exponent $\delta$ of the stream does not exceed 2 and that the affine laminations are never flat except for several explicit special cases (rational functions with parabolic Thurston orbifold).

Product Details

ISBN-13:
9780821836156
Publisher:
American Mathematical Society
Publication date:
01/03/2005
Series:
Memoirs of the American Mathematical Society Series, #820
Pages:
119
Product dimensions:
6.94(w) x 9.96(h) x 0.31(d)

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