Contractions isolate maps that pull points uniformly closer together. In a [metric space](/page/Metric%20Space), continuity only says that nearby points remain nearby after applying a map; a contraction imposes a quantitative improvement on every distance at once. This stronger condition is the engine behind fixed point arguments, iterative algorithms, and stability estimates in [complete metric spaces](/page/Complete%20Metric%20Space), [Banach spaces](/page/Banach%20Space), and differential equations.

The word also appears in other areas, such as contraction of a loop, contraction of a matroid, contraction under localization, and contraction as a structural rule in [proof theory](/page/Proof%20Theory). Those are separate constructions with their own ambient categories. On this page, the unqualified analytic meaning is the metric-space notion: a self-map whose Lipschitz constant is strictly less than $1$.

## Definition

The basic problem is to formalise the idea that applying a map makes all configurations smaller by a uniform factor. A map may decrease some distances and enlarge others; it may even bring a single orbit closer to a point while stretching nearby points. The contraction condition rules out such local accidents by requiring one constant that works for every pair of points.

[definition: Contraction] Let $(X,d)$ be a metric space. A map $f: X \to X$ is a contraction if there exists a constant $c \in [0,1)$ such that \begin{align*} d(f(x),f(y)) &\le c\,d(x,y) \end{align*} for all $x,y \in X$. [/definition]

A first test case is a map that simply scales all distances by the same factor. This gives the reader a concrete model before the general fixed point machinery enters.

[example: Basic Scaling Contraction] Let $X=\mathbb{R}$ with the usual metric $d(x,y)=|x-y|$, and define $f:\mathbb{R}\to\mathbb{R}$ by $f(x)=x/2$. For arbitrary $x,y\in\mathbb{R}$, we have \begin{align*} d(f(x),f(y)) &= |f(x)-f(y)| \end{align*} and substituting the formula for $f$ gives \begin{align*} |f(x)-f(y)| &= \left|\frac{x}{2}-\frac{y}{2}\right|. \end{align*} Factoring out $1/2$ inside the absolute value, \begin{align*} \left|\frac{x}{2}-\frac{y}{2}\right| &= \left|\frac{x-y}{2}\right|=\frac{1}{2}|x-y|. \end{align*} Since $1/2\in[0,1)$, the inequality required in the definition holds with equality: \begin{align*} d(f(x),f(y)) &= \frac{1}{2}d(x,y). \end{align*} Thus $f$ is a contraction with contraction constant $1/2$; in this model case every distance is exactly halved. [/example]

The constant $c$ is not part of the map's data unless it is explicitly specified. What matters is the existence of some uniform factor strictly below $1$. To compare maps quantitatively, it is useful to record the smallest global distance factor allowed by the metric estimate.

[definition: Lipschitz Constant] Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, and let $f: X \to Y$ be a map. The Lipschitz constant of $f$ is \begin{align*} \operatorname{Lip}(f) &= \inf\{L \ge 0 : d_Y(f(x),f(y)) \le L d_X(x,y) \text{ for all } x,y \in X\}. \end{align*} [/definition]

With this language, a contraction is a self-map admitting a Lipschitz bound with a constant strictly below $1$. A nearby boundary case is important enough to name: maps that never increase distance, but may preserve it, occur naturally in geometry and dynamics.

[definition: Nonexpansive Map] Let $(X,d)$ be a metric space. A map $f: X \to X$ is nonexpansive if \begin{align*} d(f(x),f(y)) &\le d(x,y) \end{align*} for all $x,y \in X$. [/definition]

Every contraction is nonexpansive, but the converse fails. The identity map on any metric space with at least two points is nonexpansive and is not a contraction. The main payoff of the strict inequality is that iteration is forced toward a single possible endpoint, so the target endpoint needs its own term.

[definition: Fixed Point] Let $X$ be a set and let $f: X \to X$ be a map. A point $x^* \in X$ is a fixed point of $f$ if \begin{align*} f(x^*) &= x^*. \end{align*} [/definition]

Fixed points convert equations into geometry. Solving $f(x)=x$ can be easier than solving an equation directly, because repeated application of a contraction forces all starting points into the same limiting location when the ambient space is complete.

## Equivalent Characterisations

For computations, it is common to verify contraction through a supremum of distance ratios. This turns a universal inequality over all pairs into a single numerical bound. It is useful when estimates are obtained by comparing output distances with input distances directly.

[quotetheorem:9874]

The supremum condition is not just a convenient test; it is the sharp way to measure the best possible contraction constant. If the supremum is below $1$, one constant works uniformly for every pair of distinct points. If it is equal to or above $1$, the map may still shrink some distances, but the contraction argument has no global ratio strong enough to force convergence from arbitrary starting points.

Many analytic settings do not begin with an abstract distance formula; they begin with a norm and estimates on differences of vectors or functions. To use the distance-ratio criterion there, we need the same idea expressed in terms of norms of differences, because those are the quantities that estimates in vector and function spaces naturally control.

[quotetheorem:9875]