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 $\sum |b_j - 1| < \infty$

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    $\begingroup$ Knowing nothing about this, it seems like a "time-inhomogeneous" dynamical system, and one can transform it into a traditional dynamical system by enlarging the state space, the same way one does for Markov chains. Namely, define $\Phi : \mathbb{N} \times G \to \mathbb{N} \times G$ by $\Phi(n,x) = (n+1, \phi_n(x))$. Then your $\mho$ are just the iterates of $\Phi$. $\endgroup$ Feb 21, 2017 at 5:36
  • $\begingroup$ I'll have to search up "time-inhomogeneous dynamical system". I'm assuming the problem at hand involves some kind of dynamics, so I'll give it a shot. Though "time-inhomogeneous dynamical system" sounds like some kind of boogey man professors tell their grad students about. $\endgroup$
    – user78249
    Feb 21, 2017 at 6:57
  • $\begingroup$ It's not a real term for this as far as I know; I just made it up, by analogy with time-inhomogeneous Markov chains. I also tried searching for "time-inhomogeneous dynamical system" but it seems to refer to something else. So if this has been considered, it's under some other name. $\endgroup$ Feb 21, 2017 at 14:04

1 Answer 1


Go to researchgate and view a number of research notes I have written on the subject. Also, visit "Infinite Compositions of Analytic Functions" on Wikipedia. Much of the elementary theory can be found in both locations, as well as theorems concerning holomorphic functions only. Frequently, Lipshitz contraction suffices for non-holomorphic functions, but for the analytic case a kind of domain contraction does the job. They are not necessarily the same. Proofs for the analytic cases are more complicated than those for more general complex functions where simple elementary arguments are the norm.

A notation already exists for both "Outer" or "Left" compositions and "Inner" or "Right" compositions. You'll see. There are very few mathematicians interested in this topic, which is a bit surprising since it extends the concept of iteration considerably.

And the computer imagery arising can be delightful as well!

John Gill

  • $\begingroup$ I will do just as you say. I feel like I just found a comrade in arms. I've been mostly studying infinite composition by myself; frustrated by the lack of literature. I, too, feel this is a great subject. You can make some pretty crazy functional equations by representing holomorphic functions using infinite compositions. A solution of $F(s) + e^{sF(s)} = F(s+1)$ is nearly trivial when using infinite compositions. Just another off beat 'useless' thing that appears in infinite compositions. $\endgroup$
    – user78249
    Mar 12, 2017 at 4:47
  • $\begingroup$ James, if you go to my website, johngill.net , at the bottom of the home page is an e-mail address you can use to contact me if you wish. I got started on the topic in 1970 when I wrote a paper on infinite compositions of linear fractional transformations (Trans AMS, 1973) that followed several other such papers by analytic continued fractions' investigators. $\endgroup$
    – John Gill
    Mar 12, 2017 at 22:56

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