A special case of Birkhoff's ergodic theorem for a nonexpansive map

Question

I have the following difference equation: $x_{t+1}=f(x_t)=x_t+3.61*(1/x_t-1)$ where the domain is the closed interval E=[0.19,17.48]. I know the following facts for this map.

1. This map is smooth and unimodal (convex) with the modal point at x=1.9 (and $f(1.9)=0.19$).
2. $f|_{E}$ is a map to $E$.
3. $f|_{E}$ satisfies the Li-Yorke condition: $f(3.75)=1.9, f(1.9)=0.19, f(0.19)=15.58>3.75$, so $E$ contains an uncountable scrambled set.
4. A numerical simulation suggests that $\lim_{N\rightarrow \infty}\frac{1}{N}\sum_{j=1}^N{f^{(j)}{(x)}}$ is likely to converge to 4.46 for almost all $x\in E$.

Anybody gives me any suggestion/help to prove part 4 in a rigorous way? Any comment (partial answer, reference) is welcome. For example, I would like to know how to prove $\lim_{N\rightarrow \infty}\frac{1}{N}\sum_{j=1}^N{f^{(j)}{(x)}}$ converges to a constant.