ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: X Bhattacharyya, P., Indian Institute of Technology, Madras, India On the domain of normality in iteration theory Let f be a function of the complex variable z. The natural iterates f_{n} of f are defined inductively by

f_{0}(z) = z, f_{1}(z) = f(z), f_{n+1}(z) = f(f_{n}(z)), n = 0, 1, 2, ⋯

The Fatou set F is defined as F(f) = {f_{n}(z) is normal (in the sense of Montel) in some neighbourhood of z}. Its complement is the Julia set J (f). Both F(f) and J (f) have been intensively studied in the recent years. If f is a nonlinear analytic function, then J (f) is closed, nonempty perfect and has no interior points unless it is the whole plane. J (f) divides the plane into various components of F(f). In this paper we consider some results on the components of F(f). For example we prove some sufficient conditions under which F(f) is empty for entire and meromorphic f. We also prove some theorems on the existence of fixed points on the boundary of some components of F(f).