Conformal and Harmonic Measures on Laminations Associated with Rational Maps

The framework of aane and hyperbolic laminations provides a unifying foundation for many aspects of conformal dynamics and hyperbolic geometry. The central objects of this approach are an aane Riemann surface lamination A and the associated hyperbolic 3-lamination H endowed with an action of a discrete group of iso-morphisms. This action is properly discontinuous on H, which allows one to pass to the quotient hyperbolic lamination 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 aane lamination A (analogues of the Patterson{Sullivan conformal measures for Kleinian groups), harmonic and invariant measures on the corresponding hyperbolic lamination H, the \Anosov-Sinai cocycle", the corresponding \basic cohomology class" on A (which provides an obstruction to atness), and the Busemann cocycle on H. A number of related geometric objects on laminations | in particular, the backward and forward Poincar e series and the associated critical exponents, the curvature forms and the Euler class, currents and transverse invariant measures,-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, M is a sublamination of the unit tangent bundle of a hyperbolic 3-manifold, its tranversals can be identiied 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-conformal stream on A and the corresponding-harmonic measure on M, where = (? 2). We prove that the exponent of the stream does not exceed 2 and that the aane laminations are never at except for several explicit special cases (rational functions with parabolic Thurston orbifold).