An ergodic theorem for the quasi-regular representation of the free group

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.