Herman rings in the theory of iterations of endomorphisms of C*

Abstract

The article consists of two parts. In the first we investigate the problem of the existence of cycles of doubly-connected components of the Fatou set of endomorphisms of C *. By comparison with the cases of rational and entire functions, additional characteristics are found here. Section (c) of Theorem 1 and Example 1 were independently obtained by Baker [3] and by the author [4]. The basic result of the second part of the article is the theorem on J-instability of the endomorphisms of C * whose Fatou sets contain an invariant Herman ring (Theorem 2). For rational functions this result was proved by Mane in [5]. Mate's method of proof can be transferred to the case of C =* with almost no changes, but we shall prove Theorem 2 with the help of the method of "quasiconformal surgery" (this approach is possible also in the case of rational functions). This method first appeared in the works of Douady and Hubbard. Surgery by Herman rings was first applied by Shishikura in [6].