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AuthorWalczak, Paweล{130}. author
TitleDynamics of Foliations, Groups and Pseudogroups [electronic resource] / by Paweล{130} Walczak
ImprintBasel : Birkhรคuser Basel : Imprint: Birkhรคuser, 2004
Connect tohttp://dx.doi.org/10.1007/978-3-0348-7887-6
Descript XI, 228 p. online resource

SUMMARY

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


CONTENT

1 Dynamical systems -- 1.1 Pseudogroups -- 1.2 First examples -- 1.3 Foliations, laminations and holonomy -- 1.4 Markov pseudogroups -- 1.5 Hyperbolic spaces and groups -- 2 Growth -- 2.1 Growth types -- 2.2 Growth in groups -- 2.3 Orbit growth for pseudogroups -- 2.4 Expansion growth -- 3 Entropy -- 3.1 Entropy of classical systems -- 3.2 Entropy of pseudogroups -- 3.3 Geometric entropy of foliations -- 3.4 Relating various entropies -- 3.5 Examples and constructions -- 3.6 Entropy and resiliency -- 4 Invariant measures -- 4.1 Basic definitions and facts -- 4.2 Transverse invariant measures and homology -- 4.3 Measures and orbit growth -- 4.4 Transverse invariant measures in codimension 1 -- 4.5 Vanishing entropy and invariant measures -- 4.6 Entropy, geodesic flow and invariant measures -- 4.7 Harmonic measures -- 4.8 Pattersonโ{128}{148}Sullivan measures -- 5 Hausdorff dimension -- 5.1 Definitions and basic facts -- 5.2 Julia sets -- 5.3 Dimension in foliated manifolds -- 5.4 Dimension of a hyperbolic boundary -- 5.5 Dimension of a limit set -- 6 Varia -- 6.1 Complexity growth -- 6.2 Expansive systems -- 6.3 Pseudo-orbits and pseudoleaves -- 6.4 Generic leaves


Mathematics Group theory Topological groups Lie groups Dynamics Ergodic theory Differential geometry Mathematics Group Theory and Generalizations Topological Groups Lie Groups Dynamical Systems and Ergodic Theory Differential Geometry



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