Hopf decomposition and horospheric
limit sets |

Vadim A. Kaimanovich |

By
looking at the relationship between the recurrence properties of a countable
group action with a quasi-invariant measure and the structure of its ergodic
components we establish a simple general description of the Hopf decompo-
sition of the action into the conservative and the dissipative parts in terms of the Radon-Nikodym
derivatives of the action.
As an application we prove that the conservative part of the boundary action of a discrete group of isometries of a Gromov hyperbolic space with respect to any invariant quasi-conformal stream coincides (mod 0) with the big horospheric limit set of the group. As an example we consider in more detail the Hopf decompo- sition in the simplest model case: a subgroup of a finitely generated free group acting on the boundary of the ambient group. |