Let $G$ be a countable group acting minimally by homeomorphisms on a compact Hausdorff space $X$ and $\mu$ be a probability measure on $G$ whose support generates $G$ as a semigroup.
Let $\nu$ is a $\mu$-stationary measure on $X$ such that $(X,\nu)$ is a $(G,\mu)$ boundary, ie for $\mu^{\mathbb{N}}$ almost every trajectory $g\_n$ the pushforwards $g\_1...g\_n \nu$ weakly converge to a dirac measure.


Does it follow that $\nu$ is the unique $\mu$-stationary Borel probability measure on $X$? This is the case in many geometric situations, for instance when the action is on the Gromov boundary of a hyperbolic group. In general I suspect the answer is false but cannot think of a counterexample...



Uniqueness of stationary measures for $(G,\mu)$ boundaries
A way to force a counter example is by taking $G$ to be an amenable group and choose $\mu$ such that the Furstenberg-Poisson boundary of $(G,\mu)$ is non-trivial.
Now take $X$ to be a compact model of this Furstenberg-Poisson boundary.
Then, by amenability, $X$ admits an invariant measure in addition to the $\nu$,
thus at least two different stationary measures.


