Hopf decomposition and horospheric limit sets

TL;DR

This paper provides a general description of the Hopf decomposition for group actions based on Radon--Nikodym derivatives and applies it to show the conservative part of a boundary action matches the big horospheric limit set in hyperbolic spaces.

Contribution

It introduces a simple description of the Hopf decomposition using Radon--Nikodym derivatives and relates it to horospheric limit sets in hyperbolic geometry.

Findings

The Hopf decomposition can be characterized via Radon--Nikodym derivatives.

The conservative part of boundary actions coincides with the big horospheric limit set.

The results connect ergodic theory with geometric group theory.

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.