Let $A$ be the compact annulus in the real plane with radii $2$ and $3$, and $(r,\phi)$ its polar coordinates. Define $F(r,\phi)=(r,r\phi)\ (\mod\ 2\pi)$. Is it true that, for a continuous function $g:A\to \mathbb R$, the integral $$\int_A g(F^n(x))dx$$ converges as $n\to+\infty$? ($F^n$ is the n-th iteration of $F$.)
My attempt was to prove that $F^n(x)$ is Cauchy for almost every $x$. This would allow us to apply Lebesgue's dominated convergence theorem, since $g$ must be bounded on the compact $A$. But I just can't obtain the desired result, so I'm beginning to suspect that this can be false, and the convergence (if it is true) comes from weaker properties.
Thank you in advance.