Let's consider the following map $$F(x, y) = \lim_{n \to \infty}{\frac{1}{n} \sum_{k=0}^{n-1}{f(\{x + ky \})}}$$ and $f(x) = x(1-x)$. I would like to evaluate the value of $F(x, y)$ for arbitrary given $x, y \in [0, 1]$.
The main idea is to apply Birkhoff theorem which states that:
For a given dynamical system $(X, \mathcal{F}, \mu)$, $T: X \rightarrow X$, which preserves measure, $\forall f \in L^{1}(\mu)$, $$\lim_{n \to 0}{\frac{1}{n} \sum_{k=0}^{n-1}{f(T^{k}x)}} = f^{*}(x)$$ $\int_{X}{f d \mu} = \int_{X}{f^{*} d \mu}$. Moreover, if $T$ is ergodic, then $f^{*}$ is constant.
Is it possible to make a direct application? Seems that it's reasonable to consider $T(x, y) = \{x+ky \}$, but then the next step should be looking for $\mu$ so that $T$ is $\mu$ invariant.
Are there any hints that might help? Any sort of help would be much appreciated.