## Illinois Journal of Mathematics

### Backward iteration in the unit ball

Olena Ostapyuk

#### Abstract

We will consider iteration of an analytic self-map $f$ of the unit ball in $\mathbb{C}^{N}$. Many facts were established about such dynamics in the 1-dimensional case (i.e., for self-maps of the unit disk), and we will generalize some of them in higher dimensions. In particular, in the case when $f$ is hyperbolic or elliptic, it will be shown that backward-iteration sequences with bounded hyperbolic step converge to a point on the boundary. These points will be called boundary repelling fixed points and will possess several nice properties. At each isolated boundary repelling fixed point, we will also construct a (semi) conjugation of $f$ to an automorphism via an analytic intertwining map. We will finish with some new examples.

#### Article information

Source
Illinois J. Math., Volume 55, Number 4 (2011), 1569-1602.

Dates
First available in Project Euclid: 12 July 2013

https://projecteuclid.org/euclid.ijm/1373636697

Digital Object Identifier
doi:10.1215/ijm/1373636697

Mathematical Reviews number (MathSciNet)
MR2942116

Zentralblatt MATH identifier
1269.30032

#### Citation

Ostapyuk, Olena. Backward iteration in the unit ball. Illinois J. Math. 55 (2011), no. 4, 1569--1602. doi:10.1215/ijm/1373636697. https://projecteuclid.org/euclid.ijm/1373636697

#### References

• M. Abate, Iteration theory of holomorphic maps on taut manifolds, Mediterranean Press, Rende, 1989.
• I. N. Baker and Ch. Pommerenke, On the iteration of analytic functions in a half-plane, II, J. Lond. Math. Soc. (2) 20 (1979), 255–258.
• F. Bayart, The linear fractional model on the ball, Rev. Mat. Iberoam. 24 (2008), 765–824.
• L. E. Böttcher, The principal laws of convergence of iterates and their applications to analysis, Izv. Kazan. Fiz.-Mat. Obshch. 14 (1904), 155–234.
• F. Bracci and G. Gentili, Solving the Shröder equation at the boundary in several variables, Michigan Math. J. 53 (2005), 337–356.
• F. Bracci, G. Gentili and P. Poggi-Corradini, Valiron's construction in higher dimension, Rev. Mat. Iberoam. 26 (2010), 57–76.
• F. Bracci and P. Poggi-Corradini, On Valiron's theorem, Future Trends in Geometric Function Theory. Rep. Univ. Jyväskylä Dept. Math. Stat., vol. 92, University of Jyväskylä, Jyväskylä, 2003.
• M. D. Contreras, S. Díaz-Madrigal and Ch. Pommerenke, Some remarks on the Abel equation in the unit disk, J. Lond. Math. Soc. (2) 75 (2007), 623–634.
• C. Cowen, Iteration and the solution of functional equations for functions analytic in the unit disk, Trans. Amer. Math. Soc. 265 (1981), 69–95.
• C. Cowen and B. MacCluer, Schroeder's equation in several variables, Taiwanese J. Math. 7 (2003), 129–154.
• C. Cowen and Ch. Pommerenke, Inequalities for the angular derivative of an analytic function in the unit disk, J. Lond. Math. Soc. (2) 26 (1982), 271–289.
• M. Hervé, Quelques propriétés des applications analytiques d'une boule à m dimensions dans elle-même, J. Math. Pures Appl. 42 (1963), 117–147.
• M. Jury, Valiron's theorem in the unit ball and the spectra of composition operators, J. Math. Anal. Appl. 368 (2010), 482–490.
• G. Koenigs, Recherches sur les itegrales de certaines equations fonctionelles, Ann. Sci. Ec. Norm. Super. (3) 1 (1884), 3–41.
• B. D. MacCluer, Iterates of holomorphic self-maps of the unit ball in $\mathbb{C}^n$, Michigan Mat. J. 30 (1983), 97–106.
• P. Poggi-Corradini, Backward-iteration sequences with bounded hyperbolic steps for analytic self-maps of the disk, Rev. Mat. Iberoam. 19 (2003), 943–970.
• P. Poggi-Corradini, Canonical conjugations at fixed points other than the Denjoy–Wolff point, Ann. Acad. Sci. Fenn. Math. 25 (2000), 487–499.
• P. Poggi-Corradini, Norm convergence of normalized iterates and the growth of Kœ nigs maps, Ark. Mat. 37 (1999), 171–182.
• P. Poggi-Corradini, On the uniqueness of classical semiconjugations for self-maps of the disk, Comput. Methods Funct. Theory 6 (2006), 403–421.
• Ch. Pommerenke, On the iteration of analytic functions in a half-plane, I, J. Lond. Math. Soc. (2) 19 (1979), 439–447.
• S. Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math. 79 (1957), 809–824.
• G. Valiron, Sur l'itérations des fonctions holomorphes dans un demi-plan, Bull. Sci. Math. (2) 55 (1931), 105–128.