zbMATH — the first resource for mathematics

A dichotomy of Hopf for geodesic flows associated with the discrete groups isometries of trees. (Une dichotomie de Hopf pour les flots géodésiques associés aux groupes discrets d’isométries des arbres.) (French) Zbl 0816.58033
Let $$X$$ be a complete locally compact metric tree and $$\Gamma$$ a group of isometries acting properly on $$X$$. The authors prove the analogue of Hopf’s and Sullivan’s theorems for geodesic flows on manifolds of constant curvature in the present situation. For each triple $$(X,\Gamma,\mu)$$ there is a canonically defined geodesic flow in $$(\Omega,m)$$ (where $$m$$ corresponds to the $$\Gamma$$-conformal measure $$\mu$$ on the boundary $$\partial X$$). This flow is either conservative and ergodic or dissipative. It is conservative iff $$\mu(\Lambda_ c) = \mu(\partial X)$$ where $$\Lambda_ c$$ denotes the conical limit set and it is dissipative iff $$\mu(\Lambda_ c) = 0$$.

MSC:
 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 53D25 Geodesic flows in symplectic geometry and contact geometry 37A99 Ergodic theory 28D10 One-parameter continuous families of measure-preserving transformations 37D99 Dynamical systems with hyperbolic behavior 37C10 Dynamics induced by flows and semiflows
Full Text: