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Foliations 1
Foliations 1
Date: 28 January 2011, 12:39

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This is the first of two volumes on the qualitative theory of foliations. It is divided into three parts, the first being a primer on foliated manifolds. The material here could be the basis of a short graduate course or learning seminar, preparing students to begin exploring the literature. The remainder of this volume is a "sampler" of more advanced topics. In Part 2, we focus on foliated manifolds of codimension one. Here, hands-on geometric methods yield a structure theory (the theory of levels) which is reminiscent of the classical Poincare-Bendixson theory of flows on surfaces. In arbitrary codimension (Part 3) these methods break down and we turn to the techniques of ergodic theory. Again the theory of flows, now in higher-dimensional manifolds, is the model. Specifically, we consider foliations to be generalized dynamical systems and extend to these systems the theory of invariant measures (following J. F. Plante [107], D. Sullivan [127], et al) and the theory of topological entropy (following E. Ghys, R. Langevin and P. Walczak [50]). For the most part, these methods apply equally well to laminations (partial foliations of manifolds) and even to abstract laminations (foliated metric spaces). In Chapter 11, we introduce this abstract setting and most of the subsequent results are proven for foliated metric spaces.


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