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Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces. (English) Zbl 0803.58032

The subject of the paper is the Hopf alternative between ergodicity and complete dissipativity for invariant measures of the geodesic flow on a quotient \(X = \widetilde{X}/G\) of a Gromov hyperbolic space \(\widetilde{X}\) (particular case: \(X\) is a Riemannian manifold with pinched negative curvature, \(G = \pi_ 1(X)\), and \(\widetilde{X}\) is the universal covering space of \(X\)). We prove it for a wide class of invariant measures corresponding to quasi-product geodesic currents. The method of the proof is new and relies on using induced transformations instead of the ratio ergodic theorem.
A \(G\)-invariant Markov operator \(\widetilde{P}\) on \(\widetilde{X}\) determines in a natural way an invariant measure of the geodesic flow on \(X\). The main result of the paper is that either the geodesic flow is ergodic with respect to the harmonic invariant measure and the quotient operator \(P\) on \(X\) is recurrent, or it is completely dissipative and the quotient operator is transient. In the particular case when \(X\) has constant negative curvature, and \(P\) is the Markov operator of the Brownian motion on \(X\) this result has been known as the Hopf-Tsuji- Sullivan theorem. Our proof is based on the theory of Markov operators and does not use neither the Birkhoff ergodic theorem nor the general Hopf dichotomy. Applications to conformal densities and to discrete hyperbolic spaces (hyperbolic groups, covering trees) are discussed.
Reviewer: V.A.Kaimanovich

MSC:

37A99 Ergodic theory
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
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