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

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

2003-08-06

arxiv.org/abs/math/0308060v1

Mathematics

Complex Variables

32 pages, to appear in Revista Matematica Iberoamericana

Scientific paper

A lot is known about the forward iterates of an analytic function which is bounded by 1 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.

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