Ergodicity of Harmonic Invariant Measures for the Geodesic Flow on Hyperbolic Spaces

The notions of ergodicity (absence of non-trivial invariant sets) and conservativity (absence of non-trivial wandering sets) are basic for the theory of measure preserving transformations. Ergodicity implies conservativity, but the converse is not true in general. Nonetheless, transformations from some classes always happen to be either ergodic (hence, conservative), or completely dissipative (i.e., their ergodic components are just orbits in the state space). The rst statement of this type was proved by Hopf Ho] for the geodesic ow on Riemannian surfaces with constant negative curvature (whence the name \the Hopf dichotomy"). It was generalized by Sullivan to the geodesic ow on higher dimensional Riemannian manifolds with constant negative curvature Su2] (both Hopf and Sullivan considered the Liouville invariant measure of the geodesic ow). It turned out that ergodicity of the geodesic ow is equivalent to recurrence of the Brown-ian motion on the manifold (in this form it is called the Hopf{Tsuji{Sullivan theorem). Later Sullivan generalized this result to invariant measures arising from a conformal density on the sphere at innnity Su3] (see also a survey in N] and a recent paper Y]). Of course, the Liouville measure is not the only invariant measure of the geodesic ow. Another natural invariant measure of the geodesic ow on a negatively curved Riemannian manifold with pinched sectional curvatures is the harmonic invariant measure connected with the Brownian motion (see K1]). The harmonic invariant measure coincides with the Liouville measure in the constant curvature case, or, more generally, for rank 1 locally symmetric manifolds. Under some additional assumptions, the harmonic invariant measure can be associated with any Markov operator on a Gromov hyperbolic space. The aim of this paper is to prove an analogue of the Hopf dichotomy for the harmonic measure in this generality: either the quotient Markov operator is recurrent and the geodesic ow is ergodic with respect to the harmonic invariant measure, or the quotient operator is transient and the geodesic ow is completely dissipative. Although the general scheme of our proof is the same as in Sullivan's papers Su2], Su3], an auxiliary intermediate condition being that the harmonic measure is concentrated on the radial limit set, the presentation and the details are diierent.