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Ergodic theory of foliations and a theorem of Sacksteder

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Dynamical Systems

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1342))

Abstract

We introduce the leafwise geodesic flow of a foliation, a flow on the unit tangent bundle to the leaves which preserves the natural foliation on this manifold. The transverse dynamics of this flow closely mirror the dynamics of the original foliation, and in this paper we outline a program for the study of foliation dynamics based on this observation. For example, the topological entropy of a foliation is defined to be the toplogical entropy of this flow relative to the invariant foliation. This yields a topological entropy close to that defined by Ghys-Langevin-Walczak. The metric entropies of a foliation are defined to be the corresponding relative metric entropies of the leafwise geodesic flow, with respect to invariant measures for the flow. The topological entropy then dominates the metric entropies, and the supremum of the metric entropies over the space of probability measures equals the topological entropy. This extends to foliations the relative variational principle of Ledrappier and Walters. Upper estimates of foliation metric entropies via transverse Lyapunov exponents are given, extending work of Strelcyn, from which we deduce a generalization of a theorem of Sacksteder concerning the existence of linearly contracting holonomy in exceptional minimal sets for codimension-one foliations of differentiability class Holder C 1.

This work was supported in part by a Grant from the Sloan Foundation and the NSF. The hospitality of the Mathematical Institute at Oxford and the I.H.E.S. is gratefully acknowledged.

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Authors and Affiliations

  1. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 60680, Chicago, Illinois, USA

    Steven Hurder

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  1. Steven Hurder
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James C. Alexander

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© 1988 Springer-Verlag

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Hurder, S. (1988). Ergodic theory of foliations and a theorem of Sacksteder. In: Alexander, J.C. (eds) Dynamical Systems. Lecture Notes in Mathematics, vol 1342. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0082838

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  • DOI: https://doi.org/10.1007/BFb0082838

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