Let $A \subset \mathbb{R}^2$ be the annulus defined, in polar coordinates, as

$$ A = \{ (r, \theta) \in [0, \infty) \times [0, 2\pi) : 2 \leq r \leq 3 \} \, .$$
Let $F: A \to A$, $F:(r, \theta) \mapsto (r, r \theta \text{ (mod } 2\pi))$.
Let $g : A \to \mathbb{R}$ be a continuous function. Discuss the existence of the limit
$$ \lim\_{n \to \infty} \int\_A g \circ F^{(n)}(x) \, \text{d}x \, ,$$
where $F^{(n)} = F \circ F \circ \dots \circ F$ denotes the composition of $F$ with itself $n$ times.

I cannot find a way to attack this problem. Here's what I tried: if $g \circ F^{(n)}$ converged, I think we could take the limit under the integral sign. So I tried to show the convergence of $F^{(n)}$. If it converged, then the limit function must send every point to a fixed point of $F$. So I determined the fixed points of $F$. But I don't think $F^{(n)}$ converges.