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Item Details
| Title:
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CONFORMAL AND HARMONIC MEASURES ON LAMINATIONS ASSOCIATED WITH RATIONAL MAPS
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| By: |
Vadim A. Kaimanovich, Mikhail Lyubich |
| Format: |
Paperback |
| List price:
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£66.95 |
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We currently do not stock this item, please contact the publisher directly for
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| ISBN 10: |
0821836153 |
| ISBN 13: |
9780821836156 |
| Publisher: |
AMERICAN MATHEMATICAL SOCIETY |
| Pub. date: |
15 January, 2005 |
| Series: |
Memoirs of the American Mathematical Society No. 173 |
| Pages: |
119 |
| Description: |
Dedicated to Dennis Sullivan on the occasion of his 60th birthday, this title begins with a self-contained introduction to the measure theory on laminations by discussing the relationship between leafwise, transverse and global measures. |
| Synopsis: |
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). |
| Publication: |
US |
| Imprint: |
American Mathematical Society |
| Returns: |
Returnable |
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