Memoirs of the American Mathematical Society 2005; 119 pp; softcover Volume: 173 ISBN10: 0821836153 ISBN13: 9780821836156 List Price: US$68 Individual Members: US$40.80 Institutional Members: US$54.40 Order Code: MEMO/173/820
 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 3lamination \(\mathcal H\) 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 selfcontained 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 PattersonSullivan conformal measures for Kleinian groups), harmonic and invariant measures on the corresponding hyperbolic lamination \(\mathcal H\), the "AnosovSinai 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 3manifold, its transversals can be identified with the limit set of the Kleinian group, and we show how the classical theory of PattersonSullivan 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 \(\lambda=\delta(\delta2)\). 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). Readership Graduate students and research mathematicians interested in dynamical systems, ergodic theory, manifolds, and cell complexes. Table of Contents  Introduction
 Affine and hyperbolic laminations
 Measures and currents on laminations
 Laminations associated with rational maps
 Measures on laminations associated with rational maps
 Appendix A. Laminations associated with Kleinian groups
 List of notations
 Bibliography
