kiinteäpisteen

Kiinteäpisteen is a Finnish term that translates to "fixed point" in English. In mathematics, a fixed point of a function is an element of the function's domain that maps to itself. For a function $f: X \to X$, a point $x \in X$ is a fixed point if $f(x) = x$. The set of all fixed points of a function is called its fixed-point set.

The concept of fixed points is fundamental in various areas of mathematics. For example, in the study of iterative processes, understanding fixed points helps in analyzing the long-term behavior of a system. If a sequence generated by repeatedly applying a function converges, it often converges to a fixed point of that function.

Fixed-point theorems are a class of theorems that guarantee the existence of fixed points under certain conditions. The most famous example is the Brouwer fixed-point theorem, which states that any continuous function from a compact, convex subset of Euclidean space to itself must have at least one fixed point. The Banach fixed-point theorem, also known as the contraction mapping principle, provides a condition for the existence and uniqueness of a fixed point in a complete metric space.

Fixed points also appear in computer science, particularly in the theory of computation and program verification. They are used to define the semantics of recursive programs and to prove properties about their behavior. In economics, fixed points can represent equilibrium states in various models. The study of dynamical systems extensively uses the concept of fixed points to identify equilibrium states and to analyze their stability.