Let f be a holomorphic self-map of the unit ball in dimension 2, which does not have an interior fixed point. Suppose that f has a Wolff point p with the boundary dilatation coefficient equal to 1 and the non-tangential differential dfp = id. The local behaviours of f near p are quite diverse, and we give a detailed study in many typical cases. As a byproduct, we give a dynamical interpretation of the Burns–Krantz rigidity theorem. Note also that similar results hold on two-dimensional contractible smoothly bounded strongly pseudoconvex domains.

It is shown that the unit ball in ${{\mathbb{C}}^{n}}$ is the only complexmanifold that can universally cover both Stein and non-Stein strictly pseudoconvex domains.

We apply the methods developed in our previous work to study holomorphic functions of slow growth on coverings of pseudoconvex domains in Stein manifolds. In particular, we extend and strengthen certain results of Gromov, Henkin and Shubin on holomorphic $L^{2}$ functions on coverings of pseudoconvex manifolds in the case of coverings of Stein manifolds.

If D ⊂ ℂn is a pseudoconvex domain and X ⊂ D a closed analytic subset, the famous theorem B of Cartan-Serre asserts, that the restriction operator r : (D) → (X) mapping each function F to its restriction F|X is surjective. A very important question of modern complex analysis is to ask what happens to this result if certain growth conditions for the holomorphic functions on D and on X are added.