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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Une dichotomie de Hopf pour les flots géodésiques associés aux groupes discrets d'isométries des arbres

Authors: M. Coornaert and A. Papadopoulos
Journal: Trans. Amer. Math. Soc. 343 (1994), 883-898
MSC: Primary 58F17; Secondary 57M60, 58F03, 58F11
MathSciNet review: 1207579
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Abstract: Let X be a complete locally compact metric tree and $ \Gamma $ a group of isometries of X acting properly on this space. The space of bi-infinite geodesics in X constitutes a space GX on which $ \Gamma $ acts properly. Let $ \Omega $ be the quotient of GX by this action. The geodesic flow associated to $ \Gamma $ is the flow on $ \Omega $ which is the quotient of the geodesic flow on GX, defined by the time-shift on geodesics. To any $ \Gamma $-conformal measure on the boundary $ \partial X$ there is an associated measure m on $ \Omega $ which is invariant by the geodesic flow. We prove the following results: The geodesic flow on $ (\Omega ,m)$ is either conservative or dissipative. If it is conservative, then it is ergodic, If it is dissipative, then it is not ergodic unless it is measurably conjugate to the action of $ \mathbb{R}$ on itself by conjugation. We prove also a dichotomy in terms of the conical limit set $ {\Lambda _c} \subset \partial X$ of $ \Gamma $: the flow on $ (\Omega ,m)$ is conservative if and only if $ \mu ({\Lambda _c}) = \mu (\partial X)$, and it is dissipative if and only if $ \mu ({\Lambda _c}) = 0$. The results are analogous to results of E. Hopf and D. Sullivan in the case of Riemannian manifolds of constant negative curvature.

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