An unlikely link between the most simple branching process and the fractals of Julia sets
The limiting conditional non-extinct mean of the simple Galton Watson process can be written 1/A with A(p) varying with p. And it can be shown that A = 2\phi(1/2) where \phi is the Koenigs function for the logistic map. This function happens top plot Julia sets in the complex plane based on whether it converges or diverges when iterated.
The Galton Watson process is one of the most fundamental branching models, and branching is one of the most fundamental characteristics of life. This chapter dives deep into one of it's measures that has little coverage in the literature: The non-extinct limiting conditional mean of a sub-critical standaard Galton Watson process. Basically, if the process is still alive, what is its expected value. Towards late time, that mean tends to a constant -> 1/A.
We wanted to find a formula for A(p) in terms of p. We tried several different approaches, leaving it aside for a while before picking it up to try something new. Months went by with no luck, until eventually, with the help of Chat GPT, it was determined that no such closed formula for A(p) exists. The reason is essentially as follows: 1. A(p) can defined in terms of a limit on the survival probability of the Galton Watson process. The survival probability satisfies a quadratic iterated map which is equivalent to the logistic map with a change of variables. Because of this relationship, it can be shown that A(p) can be written in terms of the Koenigs linearisation function for the logistic map. Then, it can be proven that this function is hypertrancendental in the relevant parameter range. Meaning, there is no way of expressing it in a finite number of terms with p.