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

Source: Rev. Mat. Iberoamericana Volume 19, Number 3 (2003), 943-970.

#### Abstract

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.

First Page:
Primary Subjects: 30D05, 30D50, 39B32
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.rmi/1077293811
Mathematical Reviews number (MathSciNet): MR2053569
Zentralblatt MATH identifier: 02109872

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