contraction mapping

A contraction mapping pulls every pair of points closer together by at least a fixed proportion. Apply it once and any two points end up nearer than they were; apply it repeatedly and the whole space is steadily squeezed toward a single spot. It is a map that cannot help but bring things together.

Formally, a map T from a metric space (X, d) to itself is a contraction if there is a constant k with 0 <= k < 1 such that d(T(x), T(y)) <= k * d(x, y) for all x, y in X. The constant k is the contraction factor; it must be strictly less than 1. The strict inequality is essential — a map that merely satisfies d(T(x), T(y)) < d(x, y) without a uniform factor below 1 need not have the same powerful consequences.

Every contraction is automatically continuous, indeed Lipschitz continuous with constant k. Its real power appears on complete spaces: there it must have exactly one fixed point, and iterating the map from any starting point drives you straight to it. This is the content of the Banach fixed-point theorem.