Contents
Introduction 1
Chapter 1. Affine and hyperbolic laminations 7
1.1. Afflne plane and hyperbolic space 7
1.2. The notion of lamination 20
1.3. Cohomology of an affine lamination 24
Chapter 2. Measures and currents on laminations 27
2.1. Measures on general laminations 27
2.2. Measures and streams on affine and hyperbolic laminations 33
2.3. Conformal streams, harmonic measures and harmonic functions 39
2.4. Measures and streams on quotient laminations 43
Chapter 3. Laminations associated with rational maps 45
3.1. Construction of the affine lamination 45
3.2. The Busemann and basic cocycles of a rational map 54
3.3. An example of a special section 56
3.4. Dual basic cocycle 59
3.5. Euclidean laminations 61
Chapter 4. Measures on laminations associated with rational maps 65
4.1. The balanced measures 65
4.2. Equidistribution of leaves 69
4.3. Critical exponent 74
4.4. Transverse conformal stream and A-harmonic measure 81
4.5. Leafwise conformal streams 84
4.6. Sullivan's Riemann surface laminations 87
4.7. Problems 88
Appendix A. Laminations associated with Kleinian groups 89
A.l. Foliations associated with the hyperbolic space 89
A.2. Laminations associated with Kleinian groups 92
A.3. Metrics on the Riemann sphere 94
A.4. Conformal streams and invariant measures of the geodesic flow 97
A.5. New lines in the dictionary 102
A.6. An example of a non-Euclidean affine foliation 103
List of notations 111
Bibliography 117
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