Periodic points for actions of tori in Stein manifolds

Abstract

In one complex variable dynamics, Sullivan’s theorem ([6]) gives a complete classification of the Fatou components that can appear for a rational map f . Consequently we can have only periodic components U and U can be one of the following: 1. U attracting basin; 2. U parabolic domain; 3. U Siegel disk; 4. U Herman ring. Cases 3. and 4. are called rotation domains. In these cases, the rational map f is conjugated on U to an irrational rotation, hence by taking all the iterates of f and their limits we obtain an S1-action on U which has at most one fixed (periodic) point in U . The goal of this paper is to give conditions when the generalization of the one variable situation is true. In particular we consider actions of tori on Stein manifolds and study their periodic (fixed) points.