X},\,{\omega}\in{\partial}{\mathcal X}$$ for certain constants $C\ge 0, D>0$. We call such a stream *quasi-conformal of dimension $D$*. It is *invariant* if $g{\lambda}_x={\lambda}_{gx}$ for any $g\in G,x\in{\mathcal X}$.

In the CAT($-1$) case any invariant quasi-conformal stream is equivalent (with uniformly bounded Radon–Nikodym derivatives) to an invariant conformal *stream* of the same dimension, i.e., such that the logarithms of the Radon–Nikodym derivatives are precisely proportional to the Busemann cocycles. Given a reference point $o\in{\mathcal X}$, an invariant (quasi-)conformal stream is uniquely determined just by a *(quasi-)conformal* measure ${\lambda}_o$ with the property that $\log dg{\lambda}_o/d{\lambda}_o({\omega})$ is proportional to ${\beta}_{\omega}(go,o)$ (up to a uniformly bounded additive error).

Coornaert [@Coornaert93] proved (applying the construction used by Patterson [@Patterson76] in the case of Fuchsian groups) that for any discrete group of isometries $G$ of a Gromov hyperbolic space ${\mathcal X}$ with a finite critical exponent $D$ there exists an invariant quasi-conformal stream of dimension $D$ supported by the limit set of $G$.

It is with respect to the measure class of an invariant quasi-conformal stream ${\lambda}=\{{\lambda}_x\}$ that we study the Hopf decomposition of the boundary action. The geometric description ($**$) of the Radon–Nikodym derivatives in combination with criterion ($*$) immediately implies that the conservative part of 