Random Iteration Of Holomorphic Self-maps On Non-smooth Bounded Convex Domain

The iteration theory of complex analytic maps is a part of complex dynamic system. This paper mainly studies the convergence problem for the holomorphic maps fn∈Hol(D, D),n=1,2…, Gn=f1of2o…o fn on the bounded convex domain D of Cn.In this paper, we research random iterative convergence problem based on the classical Wolff-Denjoy theorem. As D and F meet what conditions,{Gn}=f1o… o fn is conver-gent. First, this paper studies the bounded strictly convex domain. On the bounded strictly convex domain, the consistency convergence condition is as the following:Let z0∈D and ｛Gn(z0):n≥1}∩(?)D≠(?), then any composition sequence of F convergences locally uni-formly in D to a constant,(the Theorem3.2.1).We prove the theorem according to the method of the convex set separation theorem, we give the corollary by the Theorem3.2.1. Second, This paper research on the bounded convex domain, we add consistency convergence condition as the following:Let F C Hol(D,D),and let Fo be the semi-group generated by the finite compo-sition of self-maps in F. Then for any m(1≤m≤n) dimension affine subset E C D and (?)g∈F0,g｜E≠id, g\E≠id. With this condition, We give the theorem about the convergence problem for the holomorphic maps fn€Hol(D,D), n=1,2,…, Gn=f1o…ofn (the Theorem3.2.2). We prove the theorem according to the method of Abate and Beardon, the Montel theorem and soAt the end of the paper, with the similar techniques of the main theorems we give a short proof of the Wolff-Denjoy theorem, and we explain the importance of the conditions of the theo-...