Institute for Mathematical Sciences

Preprint ims01-05

V. Kaimanovich and M. Lyubich
Conformal and Harmonic Measures on Laminations Associated with Rational Maps.

Abstract: In this work we continue the exploration of affine and hyperbolic laminations associated with rational maps, which were introduced in \cite{LM}. Our main goal is to construct natural geometric measures on these laminations: transverse conformal measures on the affine laminations and harmonic measures on the hyperbolic laminations. The exponent $\de$ of the transverse conformal measure does not exceed 2, and is related to the eigenvalue of the harmonic measure by the formula $\la=\de(\de-2)$. In the course of the construction we introduce a number of geometric objects on the laminations: the basic cohomology class of an affine lamination (an obstruction to flatness), leafwise and transverse conformal streams, the backward and forward Poincar\'e series and the associated critical exponents. We discuss their relations to the Busemann and the Anosov--Sinai cocycles, the curvature form, currents and transverse invariant measures, $\la$-harmonic functions, Patterson--Sullivan and Margulis measures, etc. We also prove that the dynamical laminations in question are never flat except for several explicit special cases (rational functions with parabolic Thurston orbifold).
View ims01-05 (PDF format)