We study the dynamics of a holomorphic self-map f of complex
projective space of degree d>1 by utilizing the notion of a
Fatou map, introduced originally by Ueda (1997) and independently
by the author (2000). A Fatou map is intuitively like an analytic
subvariety on which the dynamics of f are a normal family (such as a local stable manifold of a hyperbolic periodic point). We show that global stable manifolds of hyperbolic fixed points are given by Fatou maps. We further show that they are necessarily Kobayashi hyperbolic and are always ramified by f (and therefore any hyperbolic periodic point attracts a point of the
critical set of f). We also show that Fatou components are
hyperbolically embedded in ℙn and that a Fatou component which is attracted to a taut subset of itself is
necessarily taut.