The field of holomorphic dynamics consists of at least three closely related
• 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
. The quotient M — H
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