Mikhail Lyubich

Vadim A. Kaimanovich

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 3-lamination $\mathcal H$ 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 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 affine lamination $\mathcal A$ (analogues of the Patterson-Sullivan conformal measures for Kleinian groups), harmonic and invariant measures on the corresponding hyperbolic lamination $\mathcal H$, the 'Anosov-Sinai 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 Poincare 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 3-manifold, its transversals can be identified 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 $\delta$-conformal stream on $\mathcal A$ and the corresponding $\lambda$-harmonic measure on $\mathcal M$, where $\lambda=\delta(\delta-2)$. 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).

Conformal and Harmonic Measures on Laminations Associated with Rational Maps

Leon Armenovich Takhtadzhi͡an

Dennis Sullivan

Conference on Graphs and Patterns in Mathematics and Theoretical Physics (2001 Stony Bro

Conference on Graphs and Patterns in Mathematics and Theoretical Physics

The Stony Brook Conference, 'Graphs and Patterns in Mathematics and Theoretical Physics', was dedicated to Dennis Sullivan in honor of his sixtieth birthday. The event's scientific content, which was suggested by Sullivan, was largely based on mini-courses and survey lectures. The main idea was to help researchers and graduate students in mathematics and theoretical physics who encounter graphs in their research to overcome conceptual barriers. The collection begins with Sullivan's paper, 'Sigma models and string topology', which describes a background algebraic structure for the sigma model based on algebraic topology and transversality. Other contributions to the volume were organized into five sections: Feynman Diagrams, Algebraic Structures, Manifolds: Invariants and Mirror Symmetry, Combinatorial Aspects of Dynamics, and Physics.These sections, along with more research-oriented articles, contain the following surveys: 'Feynman diagrams for pedestrians and mathematicians' by M. Polyak, 'Notes on universal algebra' by A. Voronov, 'Unimodal maps and hierarchical models' by M. Yampolsky, and 'Quantum geometry in action: big bang and black holes' by A. Ashtekar. This comprehensive volume is suitable for graduate students and research mathematicians interested in graph theory and its applications in mathematics and physics.

Proceedings of the Conference on Graphs and Patterns in Mathematics and Theoretical Physics, Dedicated to Dennis Sullivan's 60th Birthday, June 14-21, 2001, Stony Brook University, Stony Brook, NY

Graphs and Patterns in Mathematics and Theoretical Physics

John Willard Milnor

This volume is based on a conference held at SUNY, Stony Brook (NY). The concepts of laminations and foliations appear in a diverse number of fields, such as topology, geometry, analytic differential equations, holomorphic dynamics, and renormalization theory. Although these areas have developed deep relations, each has developed distinct research fields with little interaction among practitioners. The conference brought together the diverse points of view of researchers from different areas. This book includes surveys and research papers reflecting the broad spectrum of themes presented at the event. Of particular interest are the articles by F. Bonahon, ``Geodesic Laminations on Surfaces'', and D. Gabai, ``Three Lectures on Foliations and Laminations on 3-manifolds'', which are based on minicourses that took place during the conference.

Proceedings of the Conference on Laminations and Foliations in Dynamics, Geometry and Topology, May 18-24, 1998, SUNY at Stony Brook

Laminations and Foliations in Dynamics, Geometry and Topology

Mikhail Lyubich

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