On Fatou maps into compact complex manifolds

We will deal with two topics about normality of the iterates of holomorphic self-maps in compact varieties. First, for a holomorphic self-map f of an irreducible compact variety Z, we show that if at least one subsequence of {f(n))(n >= 0) converges uniformly, then the full sequence {f(n)}(n >= 0) itself is a normal family and the full set of the limit maps is finite or has the structure of a compact commutative Lie group. Second, in the case when Z is non-singular, we deal with the dynamics on forward-invariant compact subsets outside the closures of the post-critical sets. We will describe the semi-repelling structure of the dynamics in terms of repelling points and neutral Fatou discs (center manifolds). In particular, in the case of holomorphic self-maps of projective spaces, we will obtain a stronger result.