Consider a compact $K$ of the complex plane of interior void, $f$ a rational fraction leaving $K$ invariant (i.e. $f(K) \subset K$), and $\mu$ a borelian probability measure supported by $K$, and ergodic for $f$. To any function $g \in L^1(\mu,K)$ we associate the function $h(z)=\int_K \frac{g(y) d\mu(y)}{z-f(y)}$
$h$ is analytical outside $K$.
Question : if $h$=0, does it imply that $g=0$ $\mu$-a.e. ? This is the case for $f(y)=y$, as can be seen for example through Runge's theorem.