The Schwarzian operator: sequences, fixed points and $N$-cycles

Abstract:

Given a function $f(z)$ that is analytic in a domain $D$, we define the classical Schwarzian derivative $\{f,z\}$ of $f(z)$, and mention some of its most useful analytic properties. We explain how the process of iterating the Schwarzian operator produces a sequence of Schwarzian derivatives, and we illustrate this process with examples. Under a suitable restriction, these sequences become $N$-cycles of Schwarzian derivatives. Some properties of functions belonging to an $N$-cycle are listed. We conclude the article with a collection of related open problems.