Foliations, the ergodic theorem and Brownian motion

Abstract

Regarding the leaves of a compact foliated Riemannian manifold as the orbits of a dynamical system, we will prove a Birkhoff ergodic theorem asserting that the spatial average of a function equals its time average along random paths in the leaves. There are several unexpected developments in the foundations. First, there always are measures on any compact foliated Riemannian manifold (M,F) with respect to which an adequate (brownian motion along the leaves) ergodic theorem holds. Second, relative to these measures, which are called harmonic, there is a Liouville-type theorem for any bounded Bore1 function on the manifold which is harmonic on each leaf. Namely, for almost all leaves relative to any one of these harmonic measures, this leaf harmonic function is constant on each leaf. Third, the harmonic measures have a very reasonable local characterization. In any foliation coordinate system, they are transversal sums of h(L) dx(L), where h(L) is a positive harmonic function on the leaf L and dx(L) is the Riemannian measure on L. These statements comprise Theorem 1 which is stated more precisely later in the Introduction. The first fact is surprising because up to now the only measures attached to foliations were the holonomy invariant transverse measures of Plante [ 141 and Ruelle-Sullivan [ 151 and some foliations do not admit these. The second fact is unexpected because there are foliations where all the leaves are hyperbolic planes and each hyperbolic plane, being conformal to the disk, admits 285 0022-1236183 $3.00