Series of Reciprocal of Iterated Functions

Question

Have infinite series of the form $$I(f;z)=\sum_{n=1}^{\infty}\frac{1}{f^n(z)}$$ where $$f^n=\underbrace{f\circ f\circ\dots\circ f}_{n\text{ times}}$$ been studied? One interesting example of this that spurred my interest in these series is Sylvester's sequence. One way to define Sylvester's sequence is $s_0=2$ and $s_n=s_{n-1}^2-s_{n-1}+1$, and it has the known property that $$\sum_{n=0}^{\infty}\frac{1}{s_n}=1$$ This corresponds to the fact that $I(z^2-z+1; \varphi)=1$ where $\varphi$ is the golden ratio. In fact, we have the more general fact that $$I(z^2-z+1; z)=\frac{1}{z(z-1)}$$ In studying these series I have classified all rational $f$ such that $I(f;z)$ is rational in $z$, one such example being the one above. Does anyone know any way this could be used? Or know any literature that makes use of these series?

One thing I have considered with this series is that when $f$ is rational, $I(f;z)$ converges on the escaping set minus points whose orbit contains zero. Thus for those such rational $f$ for which $I(f;z)$ has a known closed form the Julia set could maybe be described by looking at where this series (or its partial sums) strays "too far" from the value it ought to converge to.