Invariant measures

Abstract

In this chapter, we consider invariant measures for groups and pseudogroups of transformations, and for foliations, in the last case defined as measures on transversals invariant under holonomy maps. The study of such measures (called transverse invariant ones) was inaugurated by J. Plante [P1] who has shown that the existence of such measures has some influence on the topology of a foliated manifold and is related to the growth types of leaves. Some of his results are discussed here, in Sections 4.2 and 4.3. Since non-trivial invariant measures need not exist, we consider also two wider classes of measures: harmonic measures on foliated Riemannian manifolds and quasi-invariant measures for Kleinian groups. In both cases, the considered measures have several interesting, very geometric, properties and reflect some dynamics of the systems, foliations and Kleinian groups. Also, in Section 4.6, we show how measures invariant under the geodesic flow of a foliation can be applied to the proof of the implication “if no resilient leaves, then vanishing entropy” for arbitrary codimension-one C¹-foliations. This interesting idea is due to Hurder, was originated at the very beginning of the theory of geometric entropy in [Hui] and developed to a final form recently in [Hu4].