2003/08/06 by Pietro Poggi-Corradini, Poggi-Corradini, Pietro
Mathematics · #30D05 #Complex Variables (math.CV) #FOS: Mathematics #math.CV #msc:30D05
paper · pdf · doi:10.48550/arxiv.math/0308060
32 pages, to appear in Revista Matematica Iberoamericana
arxiv created 2003/08/06 · arxiv updated 2009/12/01
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.