Superattracting

Superattracting is a term used in complex dynamics to describe a fixed or periodic point where the dynamics are strongly contracting under iteration. For a holomorphic function f, a fixed point z0 is superattracting if f(z0) = z0 and the derivative vanishes there: f′(z0) = 0. More generally, a periodic point z0 of period p is superattracting if the derivative of the p-th iterate vanishes at z0, i.e., (f^p)′(z0) = 0.

Locally, a superattracting point is highly contracting. In appropriate coordinates, the map near z0 is conjugate to a monomial w ↦ w^m with m ≥ 2, so the fixed point has multiplicity m. Equivalently, the zero of f(z) − z at z0 has order m ≥ 2. The immediate basin of attraction consists of points that converge rapidly to z0 under iteration.

Periodic superattracting points occur when a critical point (where f′ vanishes) lies on a periodic orbit. Such points create immediate basins that are robust under perturbations and strongly influence the local and global dynamics, including the structure of the Julia set and Fatou components.

Example: the map f(z) = z^2 has a superattracting fixed point at z = 0, since f(0) = 0 and f′(0) = 0. Points with |z| < 1 are drawn toward 0 under iteration (z^{2^n} → 0), while points with |z| > 1 tend to infinity.

Superattracting points are central in the study of holomorphic dynamics, illustrating how criticality and local multiplicity drive global behavior. They contrast with merely attracting points where f′(z0) ≠ 0 but |f′(z0)| < 1. See also Fatou set, Julia set, and complex dynamics.