information:filtering
Lipschitz maps and contractions

A Lipschitz map is one that stretches distances by at most a bounded factor. Formally, given a real number K > 0, the map f: M 1 → M 2 {\displaystyle f\,\colon M_{1}\to M_{2}} is K - Lipschitz if d 2 (f (x), f (y)) ≤ K d 1 (x, y) for all x, y ∈ M 1. {\displaystyle d_{2}(f(x),f(y))\leq Kd_{1}(x,y)\quad {\text{for all}}\quad x,y\in M_{1}.} Lipschitz maps are particularly important in metric geometry, since they provide more flexibility than distance-preserving maps, but still make essential use of the metric. For example, a curve in a metric space is rectifiable (has finite length) if and only if it has a Lipschitz reparametrization.

A 1-Lipschitz map is sometimes called a nonexpanding or metric map. Metric maps are commonly taken to be the morphisms of the category of metric spaces.

A K -Lipschitz map for K < 1 is called a contraction. The Banach fixed-point theorem states that if M is a complete metric space, then every contraction f: M → M {\displaystyle f:M\to M} admits a unique fixed point. If the metric space M is compact, the result holds for a slightly weaker condition on f: a map f: M → M {\displaystyle f:M\to M} admits a unique fixed point if d (f (x), f (y)) < d (x, y) for all x ≠ y ∈ M 1. {\displaystyle d(f(x),f(y))<d(x,y)\quad {\mbox{for all}}\quad x\neq y\in M_{1}.}