---
abstract: '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 decomposition 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.'
address: 'Mathematics, Jacobs University Bremen, Campus Ring 1, D-28759, Bremen, Germany'
author:
- 'Vadim A. Kaimanovich'
bibliography:
- 'C:/Sorted/MyTex/mine.bib'
title: Hopf decomposition and horospheric limit sets
---

[^1]

Conservativity and dissipativity are, alongside with ergodicity, the most basic notions of the ergodic theory and go back to its mechanical and thermodynamical origins. The famous *Poincaré recurrence theorem* states that any invertible transformation $T$ preserving a probability measure $m$ on a state space $X$ is *conservative* in the sense that any positive measure subset $A\subset X$ is *recurrent*, i.e., for a.e. starting point $x\in A$ the trajectory $\{T^n x\}$ eventually returns to 