Revista Matemática Iberoamericana


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Volume 19, Issue 3, 2003, pp. 943–970
DOI: 10.4171/RMI/375

Published online: 2003-12-31

Backward-iteration sequences with bounded hyperbolic steps for analytic self-maps of the disk

Pietro Poggi-Corradini[1]

(1) Kansas State University, Manhattan, USA

A lot is known about the forward iterates of an analytic function which is bounded by $1$ in modulus on the unit disk $\mathbb{D}$. 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.

Keywords: Backward-iteration, bounded steps

Poggi-Corradini Pietro: Backward-iteration sequences with bounded hyperbolic steps for analytic self-maps of the disk. Rev. Mat. Iberoam. 19 (2003), 943-970. doi: 10.4171/RMI/375