A lot is known about the forward iterates of an analytic function which is bounded by in modulus on the unit disk . The Denjoy-Wolff Theorem describes their convergence properties and several authors, from the 1880's to the 1980's, have provided conjugations which yield very precise descriptions of the dynamics. Backward-iteration sequences are of a different nature because a point could have infinitely many preimages as well as none. However, if we insist in choosing preimages that are at a finite hyperbolic distance each time, we obtain sequences which have many similarities with the forward-iteration sequences, and which also reveal more information about the map itself. In this note we try to present a complete study of backward-iteration sequences with bounded hyperbolic steps for analytic self-maps of the disk.
Cite this article
Pietro Poggi-Corradini, Backward-iteration sequences with bounded hyperbolic steps for analytic self-maps of the disk. Rev. Mat. Iberoam. 19 (2003), no. 3, pp. 943–970DOI 10.4171/RMI/375