# Conformal and Harmonic Measures on Laminations Associated with Rational Maps

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*Vadim A. Kaimanovich; Mikhail Lyubich*

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 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 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).

#### Table of Contents

# Table of Contents

## Conformal and Harmonic Measures on Laminations Associated with Rational Maps

- Contents v6 free
- Introduction 18 free
- Chapter 1. Affine and hyperbolic laminations 714 free
- Chapter 2. Measures and currents on laminations 2734
- Chapter 3. Laminations associated with rational maps 4552
- Chapter 4. Measures on laminations associated with rational maps 6572
- Appendix A. Laminations associated with Kleinian groups 8996
- A.1. Foliations associated with the hyperbolic space 8996
- A.2. Laminations associated with Kleinian groups 9299
- A.3. Metrics on the Riemann sphere 94101
- A.4. Conformal streams and invariant measures of the geodesic flow 97104
- A.5. New lines in the dictionary 102109
- A.6. An example of a non-Euclidean affine foliation 103110

- List of notations 111118
- Bibliography 117124