Iteration of Möbius transforms and continued fractions (Q810662)

Continued fractions are traditionally studied through their convergents \(S_ n(0)=s_ 1\circ s_ 2\circ...\circ s_ n(0)\) where \(s_ k(z)=a_ k/(1+z)\). In particular in connection with convergence acceleration it has become popular to study so called modifications \(S_ n(w_ n)\) with \(w_ n\) not necessarily 0. From the point of view of discrete dynamical systems it seems natural to study \(\tilde S_ n(z)=s_ n\circ...\circ s_ 2\circ s_ 1(z)\) instead, perhaps in particular when all \(s_ k\) are equal so we are studying iteration. In this paper we show that the behaviour of \(S_ n\) can be seen in a study of \(\tilde S_ n\) if \(s_ k\) are constant in k (which is of course trivial \((S_ n=\tilde S_ n))\), \(s_ k\) form a periodic sequence, or \(s_ k\) form a convergent sequence. We also think the paper is useful as a primer of continued fractions.