On Iterates of Moebius 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 ≡ K ∪ {∞ } -S so that every Moebius 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 Moebius 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.