Electronic ISBN:  9781470404215 
Product Code:  MEMO/173/820.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 173; 2005; 119 ppMSC: Primary 37; 57; Secondary 53; 58;
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 3lamination \(\mathcal H\) 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 selfcontained 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 Poincaré 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 3manifold, 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 \(\lambda=\delta(\delta2)\). 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).ReadershipGraduate students and research mathematicians interested in dynamical systems, ergodic theory, manifolds, and cell complexes.

Table of Contents

Chapters

Introduction

1. Affine and hyperbolic laminations

2. Measures and currents on laminations

3. Laminations associated with rational maps

4. Measures on laminations associated with rational maps


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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 3lamination \(\mathcal H\) 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 selfcontained 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 Poincaré 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 3manifold, 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 \(\lambda=\delta(\delta2)\). 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).
Graduate students and research mathematicians interested in dynamical systems, ergodic theory, manifolds, and cell complexes.

Chapters

Introduction

1. Affine and hyperbolic laminations

2. Measures and currents on laminations

3. Laminations associated with rational maps

4. Measures on laminations associated with rational maps