  In \cite{BAMU}, an ergodic theorem \`a la Birkhoff-von Neumann for the action
of the fundamental group of a compact negatively curved manifold on the
boundary of its universal cover is proved. A quick corollary is the
irreducibility of the associated unitary representation. These results are
generalized \cite{BOYER} to the context of convex cocompact groups of
isometries of a CAT(-1) space, using Theorem 4.1.1 of \cite{ROBLI}, with the
hypothesis of non arithmeticity of the spectrum. We prove all the analog
results in the case of the free group $\mathbb{F}_r$ of rank $r$ even if
$\mathbb{F}_r$ is not the fundamental group of a closed manifold, and may have
an arithmetic spectrum.
