Backward iteration in the unit ball.


Show simple item record Ostapyuk, Olena 2011-08-02T19:11:19Z 2011-08-02T19:11:19Z 2011-08-02
dc.description.abstract We consider iteration of an analytic self-map f of the unit ball in the N-dimensional complex space C[superscript]N. Many facts were established about such maps and their dynamics in the 1-dimensional case (i.e. for self-maps of the unit disk), and we generalize some of them in higher dimensions. In one dimension, the classical Denjoy-Wolff theorem states the convergence of forward iterates to a unique attracting fixed point, while backward iterates have much more complicated nature. However, under additional conditions (when the hyperbolic distance between two consecutive points stays bounded), backward iteration sequence converges to a point on the boundary of the unit disk, which happens to be a fixed point with multiplier greater than or equal to 1. In this paper, we explore backward-iteration sequences in higher dimension. Our main result shows that in the case when f is hyperbolic or elliptic, such sequences with bounded hyperbolic step converge to a point on the boundary, other than the Denjoy-Wolff (attracting) point. These points are called boundary repelling fixed points (BRFPs) and possess several nice properties. In particular, in the case when such points are isolated from other BRFPs, they are completely characterized as limits of backward iteration sequences. Similarly to classical results, it is also possible to construct a (semi) conjugation to an automorphism of the unit ball. However, unlike in the 1-dimensional case, not all BRFPs are isolated, and we present several counterexamples to show that. We conclude with some results in the parabolic case. en_US
dc.language.iso en_US en_US
dc.publisher Kansas State University en
dc.subject Complex analysis en_US
dc.subject Iteration en_US
dc.subject Boundary fixed points en_US
dc.title Backward iteration in the unit ball. en_US
dc.type Dissertation en_US Doctor of Philosophy en_US
dc.description.level Doctoral en_US
dc.description.department Department of Mathematics en_US
dc.description.advisor Pietro Poggi-Corradini en_US
dc.subject.umi Mathematics (0405) en_US 2011 en_US August en_US

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