Non-periodic orbit in Newton-Raphson example

Question

Let $f(z)=\frac{z^2-1}{2z}$. I want to find a point $\xi\in\mathbb{R}$ such that the $f$-orbit of $\xi$ is non-periodic and$$ \lim_{n\to\infty} \frac1n \#\{t\leq n \,:\,f^t(\xi)<0 \}=\frac13. $$ I just don't know how to approach such problem. Any ideas?