On iterates of Möbius transformations on fields

Abstract:

Let $p$ be a quadratic polynomial over a splitting field $K$, and $S$ be the set of zeros of $p$. We define an associative and commutative binary relation on $G\equiv K\cup \{\infty \}-S$ so that every Möbius transformation with fixed point set $S$ is of the form $x$ “plus” $c$ for some $c$. This permits an easy proof of Aitken acceleration as well as generalizations of known results concerning Newton’s method, the secant method, Halley’s method, and higher order methods. If $K$ is equipped with a norm, then we give necessary and sufficient conditions for the iterates of a Möbius transformation $m$ to converge (necessarily to one of its fixed points) in the norm topology. Finally, we show that if the fixed points of $m$ are distinct and the iterates of $m$ converge, then Newton’s method converges with order 2, and higher order generalizations converge accordingly.