Truncation error bounds for the composition of limit-periodic linear fractional transformations (Q5916304)

Let \(\{t_ n\}_{n=1}^ \infty\) be a sequence of linear fractional transformations \[ t_ n(w)={a_ nw+b_ n \over w+d_ n}, \] where \(\{a_ n\}\), \(\{b_ n\}\) and \(\{d_ n\}\) approach finite limits \(a\), \(b\) and \(d\), respectively. By means of this sequence we generate a new one by composition, \[ T_ n:=t_ 1 \circ t_ 2 \circ \cdots \circ t_ n \qquad \text{for } n=1,2,3,\dots. \] The following two cases are considered: Case I: \(ad \neq b\) and \(| d+u |> | d+v |\), where \(u\) and \(v\) are the two fixed points of the linear fractional transformation \(t(w):=(aw+b)/(w+d)\). Case II: \(a=b=0 \neq d\), in which case \(u:=0\). It is well known that in both these cases \(\{T_ n(u)\}\) converges to a value \(T \in \hat \mathbb{C}:=\mathbb{C} \cup \{\infty\}\). The main idea of this paper is to establish a priori truncation error bounds \(\{E_ n\}\) for \(\{T_ n(u)\}\); that is, positive values \(E_ n\) which guarantee that the unknown limit \(T\) satisfies \[ | T-T_ n(u) | \leq E_ n \qquad \text{for }n=1,2,3\dots. \] The bounds \(E_ n\) are given in terms of an algorithm by which they can be computed in some interesting cases.