fixpoints

A fixed point of a function f from a set X to itself is an element x in X such that f(x) = x. Fixed points arise in mathematics, computer science, and related fields as representations of steady states, invariants, or the outcome of recursive processes.

Examples include the real-number equation x = cos x, which has a solution approximately 0.739085...; in general, a constant function f(x) = c has a fixed point at x = c, while the identity function f(x) = x fixes every x in X.

Existence and construction: Not every function has a fixed point. In metric spaces, a contraction mapping guarantees a unique fixed point by Banach's fixed-point theorem, and iterating x_{n+1} = f(x_n) converges to it from any starting point. In discrete settings, fixed points can be found by simple iteration or by solving the equation f(x) = x.

In order theory and computer science, fixed points underpin the semantics of recursion. If f is a monotone operator on a complete lattice, the least fixed point exists and can be obtained as the limit of the increasing chain starting from the bottom element (Kleene fixed-point theorem). The greatest fixed point may be obtained by dual construction.

Applications: fixed points describe equilibria in dynamical systems and economic models, and they support recursive definitions and program semantics in programming languages. The concept also extends to multi-valued or set-valued maps in fixed-point theorems such as Kakutani.