Random iterations of holomorphic contractions in locally convex spaces and of weaker contractions in uniform spaces
The authors prove a version of the Denjoy-Wolff theorem concerning convergence of iterates of analytic functions from a domain into itself. The context here is for an unbounded domain \(G\) in a sequentially complete locally convex space \(E\), and for an arbitrary sequence \((f_n)\) of functions which map \(G\) into a domain \(D\) which is strictly inside \(G\). The result then is that under additional technical conditions the sequence of iterates \((f_1\circ\cdots\circ f_n)_n\) converges to a constant map. Several comments are made which show that the result fails without the imposition of various additional hypotheses. The authors also discuss a generalization of this result to uniform spaces, and they also give a helpful list of references to earlier work in this classical area.
convergence of iterates of analytic functions
scientific article; zbMATH DE number 1332889
scientific article