Introduction

The field of holomorphic dynamics consists of at least three closely related

branches:

• Iteration theory of rational endomorphisms of the Riemann sphere;

• The theory of Kleinian groups;

• The theory of holomorphic foliations.

The construction of Lyubich and Minsky [LM97] brings these three branches to-

gether: its input is a rational endomorphism, and the output is a hyperbolic lami-

nation analogous to the hyperbolic manifold of a Kleinian group (or rather to the

unit tangent bundle of that manifold).

The modern theory of Kleinian groups is intimately related with the 3-dimen-

sional hyperbolic geometry which provides many deep insights and powerful tools in

both ways (see Mostow [M068], Thurston [Th91], Minsky [Mi99], etc.). This rela-

tion is based on Poincare's observation that a Kleinian group G can be extended to

a discrete group of isometries of the hyperbolic space H

3

. The quotient M — H

3

/ G

is a 3-dimensional hyperbolic manifold (or rather orbifold) whose topology and ge-

ometry reflect the combinatorial and geometric properties of G.

Sullivan's dictionary between the first two branches of holomorphic dynamics

(see [Su85]) made it natural to wonder whether there exists an analogous object

associated with a rational endomorphism / : C —• C of degree d 1. Such an

object, a hyperbolic 3-dimensional (orbifold) lamination A4/, was constructed in

[LM97]. The hyperbolization (a functorial passage from dimension 2 to dimen-

sion 3) in this construction is based on an idea different from that of the "Poincare

hyperbolization" and consisting in the observation that a natural one-dimensional

fiber bundle over an arbitrary affine Riemann surface (whose fibers consist of all

conformal metrics on the tangent space to a given point) carries a canonical hyper-

bolic metric. Thus, one produces first an affine Riemann surface lamination Af

whose leaves are isomorphic to C, then a hyperbolic 3-lamination Hf by applying to

Af the hyperbolization functor, and then finally one obtains the quotient hyperbolic

lamination Mf by factorizing Hf with respect to the action of the automorphism

/ : Hf -^ (which is a natural lift of / ) .

Any Riemannian manifold is endowed with the associated volume. It is not the

case for a (leafwise) Riemannian lamination: leafwise volumes can be organized into

a global measure only in the presence of a holonomy invariant transverse measure.

To handle this problem, L. Garnett [Ga83] introduced the notion of a harmonic

measure on a Riemannian foliation, which can play the role of the Riemannian

volume on a manifold. She showed that for foliations of compact manifolds such

a measure always exists. Actually, the results of Garnett can be placed into a

more general context of the theory of Markov chains. In these terms Garnett's

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