I've recently become very intrigued by infinite compositions. To get at what I mean by the term, I'll be as explanatory as possible.
Consider a sequence of holomorphic functions $\{\phi_j\}_{j=0}^\infty$ holomorphic on some domain $G$, sending $G$ to itself. If it helps in the following, we can assume $G$ is simply connected. Let us define the two composition operators, like the summation operator $\sum$:
$$\Omega_{j=0}^n \phi_j (z)= \phi_0(\phi_1(...\phi_{n-1}(\phi_n(z))$$
$$\mho_{j=0}^n \phi_j(z) = \phi_n(\phi_{n-1}(...\phi_1(\phi_0(z))$$
where they are related by the equation
$$\Omega_{j=0}^n \phi_{n-j} = \mho_{j=0}^n \phi_j$$
Now where these objects get interesting is when we take the limit as $n\to\infty$. This produces an object not so prevalent in the literature; quite literally I've only found a few notes or a few papers. I've read that in classical analysis these objects were investigated, with little headway though. It only found a home in continued fractions where $\phi_j(z) = \frac{1}{c_j + z}$ for some sequence $\{c_j\}_{j=0}^\infty$.
I am interested in sufficient conditions for which: $f(z) = \lim_{n\to\infty} \Omega_{j=0}^n \phi_j(z)$ converges uniformly on compact subsets of $G$, and $h(z) = \lim_{n\to\infty}\mho_{j=0}^n \phi_j(z)$ converges uniformly on compact subsets of $G$. So that $f,h: G \to G$ and are holomorphic. But, I am not interested if $\phi_j(z) = \phi(z)$ is a single function for all $j$. Where this question boils into the convergence of $\phi^{\circ j} = \phi \circ \phi \circ \phi \circ ... \circ \phi$.
What is so interesting about this question, is that it requires a criterion for not only the convergence of sums, but also the convergence of products. If $\phi_j(z) = a_j + z$, then quite obviously if $\Omega_{j=0}^\infty \phi_j(z) = f(z)$, then $f(z) = z+ \sum_{j=0}^\infty a_j$. Similarly if $\phi_j(z) = b_j z$, then if $f(z) = \Omega_{j=0}^\infty \phi_j(z)$, then $f(z) = z \prod_{j=0}^\infty b_j$.
A natural decision algorithm, which applies in both of the above cases, is that the infinite composition converges uniformly if
$$\sum_{j=0}^\infty |\phi_j(z) - z| <\infty$$
Which similarly works if $\phi_j(z) = b_jz + a_j$ i.e; in the linear case. I think this condition may work for all sequences of holomorphic functions if strengthened to Normal convergence, but I am not certain. Therefore my question is in two parts.
Does there exist a reasonable criterion for uniform convergence of $\Omega_{j=0}^n \phi_j$ as $n\to\infty$? (Same question with $\mho_{j=0}^n \phi_j$.)
Secondly, if $\phi_j: G \to G$ where $\phi_j$ is holomorphic, and for all $K\subset G$ with $K$ compact, $\sum_{j=0}^\infty ||\phi_j(z) -z||_K < \infty$ does $\Omega_{j=0}^n \phi_j$ converge uniformly on $K$ to a non constant function? (Same question with $\mho_{j=0}^n\phi_j$.)
I have a rough proof of the second question, but I am skeptical of it. So any kind of counter example would be great, and really the desired answer. If it happens to be true, then a reference would be fantastic. But if you can prove it in the length it takes to write an MO post I'd be blown away as I toiled on my rough proof for a couple of months.
The intuition is really quite simple as to why the infinite composition would converge if the sum converges. Quite simply, if the infinite composition converges to a non constant function, then the tail of $\phi_j$ tends to the identity function. The important part is that if the tail converges fast enough to the identity function, the infinite composition converges. Similarly to how $\sum a_j$ converges if $a_j \to 0$ fast enough, and $\prod b_j$ converges if $b_j \to 1$ fast enough. Where both instances of fast enough are identified by the fact $\sum |a_j| < \infty$ and $\sum |b_j - 1| < \infty$
