Asymptotic Behaviour of Solutions of Linear Recurrences and Sequences of Möbius-Transformations Original Research Article

This paper is mainly concerned with the study of recurrences defined by Möbius-transformations, whose solutions are the orbits of points on the Riemann-sphere under a sequence of Möbius-transformations. We study the asymptotic behaviour of such solutions in relation to the asymptotic behaviour of the coefficients of the Möbius-transformations. Most of the theorems give sufficient conditions in order that there exist converging solutions, but a section of examples is added where examples are given of recurrences whose solutions do not converge because one or several of the conditions of the theorems are violated. One of the most important results of this paper is that if the fixpoints of the Möbius-transformations are of bounded variation and converge to distinct limits, then the behaviour of the solutions depends entirely on the products of the derivatives in the fixpoints. Several methods will be proposed to deal with the case that the fixpoints converge to one single limit. The paper starts with a few results onnth order recurrences and matrix recurrences and concludes with an investigation of the asymptotic behaviour of the solutions of linear second-order recurrences having coefficients that are asymptotic expressions in fractional powers of the indexn. A number of examples are added in order to show how some of the theorems can be applied.