Critical orbits of holomorphic maps on projective spaces (Q1307109)

In the theory of iteration of rational functions, the following theorem has a systematically interesting application. Theorem. Let \(R\) be a rational map from the Riemann sphere onto itself with degree \(\geq 2\) and suppose that the family \(\{S_n\); \(n\geq 1\}\) is such that each \(S_n\) is a single-valued analytic branch of some \((R^m)^{-1}\) (\(R^m\) is the \(m\)th iteration) in a domain \(D.\) Then \( \{S_n\); \(n\geq 1\}\) is a normal family. In this paper, the author generalizes the above theorem to maps of a complex projective space \(\mathbb{P}^n\) onto itself. The main result is the following theorem. Theorem 2.1. Let \(f:\mathbb{P}^n\rightarrow \mathbb{P}^n\) be a holomorphic map of degree \(\geq 2\) and \(h:Z\rightarrow \mathbb{P}^n\) a holomorphic map, where \(Z\) is a complex analytic space. Let \(\{g_v\}\) be a family of lifts of \(h\) by iterates of \(f.\) Then \(\{g_v\}\) is a normal family. Based on the above theorem, some interesting results are obtained as well: (1) For the above map \(f\), every Fatou limit point is contained either in a rotation domain or in the set \(E\) (Theorem 4.9). (2) If \(f:\mathbb{P}^n\rightarrow \mathbb{P}^n\) is a critical finite holomorphic map, then the Fatou limit set \(\Lambda \) is contained in the set of singular points of \(D\) (Theorem 5.4). (3) In the case of dimension 2, a sufficient condition for the Fatou set to be empty is obtained (Theorem 5.9).