Difference equations in the complex plane (Q1079752)

The difference equation \(z_{n+1}=T(z_ n)\) for \(n=0,1,2,...\), where T is an analytic function in the complex plane with fixed point \(z^*\) is studied subject to the condition \(| T'(z)| <1\) for all \(z\in {\mathbb{C}}\), \(z\neq z^*\), \(| T'(z)| <1\) for all z in a neighbourhood of \(z^*\) and \(| T'(z^*)| =1\), resp. Moreover iterations of \(C^ 1\)-maps from \({\mathbb{R}}^ n\to {\mathbb{R}}^ n\) are considered. The obtained results and applied methods are compared to theorems and proof-methods in \textit{M.-H. Shih} and \textit{C.-C. Yeh} [ibid. 81, 182-188 (1981; Zbl 0456.30034)].