Complex dynamics of the limit periodic system \(F_ n(z)=F_{n-1}(f_ n(z))\), \(f_ n\to f\) (Q2644798)

Let \(\{f_ n\}\) be a given sequence of analytic functions \(f_ n\), where the domain of each function contains the range of the next one, and where \(f_ n\to f\) uniformly on some domain S. The author continues his studies of the sequence \(\{F_ n\}\), defined by \(F_ n=F_{n-1}\circ f_ n\). The particular case where all \(f_ n\) are Möbius transformations is of importance in the analytic theory of continued fractions, and has been studied by the author and others in several papers. In the present paper the question about analytic dynamical behavior of \(\{F_ n(z)\}\) as compared to that of \(\{f^ n(z)\}\) is studied, and strong similarities are proved in several theorems. A direct extension of an \(f^ n\)-result is that \(F_ n(z)\to \lambda\) for all \(z\in S\) if f maps S into the closure of S. In the other theorems an essential condition is that \(f_ n\to f\) fast enough. One such result is a ``shadowing theorem'', stating that \(\{F_ n\}\) uniformly shadows \(\{f^ n\}\) \((| F_ n(z)-f^ n(z)| <\epsilon\) for all n and all \(z\in S)\). Tail results and results on chaotic behavior are proved. The theorems are illustrated by several examples on limit periodic continuous fractions and other limiting structures.