Dynamics of Foliations, Groups and Pseudogroups

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Springer Science & Business Media, Apr 23, 2004 - Mathematics - 228 pages
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Foliations, groups and pseudogroups are objects which are closely related via the notion of holonomy. In the 1980s they became considered as general dynamical systems. This book deals with their dynamics. Since "dynamics” is a very extensive term, we focus on some of its aspects only. Roughly speaking, we concentrate on notions and results related to different ways of measuring complexity of the systems under consideration. More precisely, we deal with different types of growth, entropies and dimensions of limiting objects. Invented in the 1980s (by E. Ghys, R. Langevin and the author) geometric entropy of a foliation is the principal object of interest among all of them.

Throughout the book, the reader will find a good number of inspirating problems related to the topics covered.

 

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Contents

Dynamical systems
1
12 First examples
3
13 Foliations laminations and holonomy
6
14 Markov pseudogroups
15
15 Hyperbolic spaces and groups
20
Growth
33
21 Growth types
34
22 Growth in groups
36
36 Entropy and resiliency
89
Invariant measures
97
42 Transverse invariant measures and homology
101
43 Measures and orbit growth
102
44 Transverse invariant measures in codimension 1
104
45 Vanishing entropy and invariant measures
106
46 Entropy geodesic flow and invariant measures
108
47 Harmonic measures
125

23 Orbit growth for pseudogroups
44
24 Expansion growth
51
Entropy
61
31 Entropy of classical systems
62
312 Invariant measures
65
313 Measuretheoretic entropy
66
314 Examples
68
315 Variational principle
70
32 Entropy of pseudogroups
73
33 Geometric entropy of foliations
76
34 Relating various entropies
79
35 Examples and constructions
84
352 Gluing
85
353 Turbulization
86
48 PattersonSullivan measures
142
Hausdorff dimension
155
52 Julia sets
161
53 Dimension in foliated manifolds
166
54 Dimension of a hyperbolic boundary
176
55 Dimension of a limit set
179
Varia
183
62 Expansive systems
187
63 Pseudoorbits and pseudoleaves
191
64 Generic leaves
201
Bibliography
211
Index
223
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