Fixed point (mathematics)

In mathematics, a fixed point is a value that remains unchanged after a specific transformation is applied. Also known as an invariant point, it represents a state of stability within a system. For a function to have a fixed point, the value must belong to both the domain and the codomain. The domain is the set of all possible input values, while the codomain is the set of potential output values. If these two sets do not overlap, the function cannot have a fixed point.

When we look at functions defined on real numbers, we can visualize fixed points using a graph. In a Euclidean plane, a function appears as a curve. A fixed point occurs wherever this curve intersects the diagonal line where y equals x. For example, if a function is defined as f(x) = x^3 - 3x^2 + 3x, the values 0, 1, and 2 are its fixed points. This is because plugging those specific numbers into the function results in the same numbers. However, not every function possesses such points. A function like f(x) = x + 1 has no fixed points because adding one ensures the output is never equal to the input.

Numerical analysis uses a specific technique called fixed-point iteration to locate these values. This process involves choosing an initial point in the domain and applying the function repeatedly. This creates a sequence of numbers where each new value is the result of the previous one. Mathematicians hope this sequence will converge, meaning it settles toward a specific value. If the function is continuous, the point it approaches is often a fixed point. Within this process, scientists categorize points as attracting, repelling, or periodic. An attracting fixed point pulls nearby values toward it, while a repelling point pushes them away.

Mathematical theorems provide rules to prove that fixed points must exist under certain conditions. The Brouwer fixed-point theorem, established in 1911, states that any continuous function from a closed unit ball in n-dimensional Euclidean space to itself must have at least one fixed point. While powerful, it does not provide a method for finding that point. In 1922, the Banach fixed-point theorem offered a different approach. It provides a criterion that guarantees fixed-point iteration will always converge to a fixed point. Other researchers, such as Lefschetz and Nielsen, developed methods in algebraic topology to actually count the number of fixed points present.

Fixed points are also vital in algebra and topology. In group theory, a point is a fixed point if a group action leaves it unchanged. This concept extends to automorphisms of rings and fields, where fixed points form specific structures like fixed subrings or fixed fields in Galois theory. In topology, a space has the fixed-point property if every continuous function on it has a fixed point. This property is a topological invariant, meaning it is preserved through homeomorphisms and retractions. While the Brouwer theorem shows that compact and convex subsets have this property, researchers like Kinoshita proved that compactness and contractibility alone are not enough to guarantee it.

In the field of order theory, the concept is expanded through partial orders. Here, mathematicians define prefixpoints and postfixpoints. A prefixpoint is any value where the function's result is less than or equal to the original value. A postfixpoint is where the original value is less than or equal to the result. A true fixed point is both a prefixpoint and a postfixpoint. Furthermore, some functions possess a least fixed point, which is the smallest fixed point according to the specific order of the set. If a monotone function exists on a complete lattice, the Knaster–Tarski theorem shows it will have a unique least fixed point.

Beyond pure math, fixed points describe stability in many real-world systems. In economics, the Nash equilibrium of a game is a fixed point of the best response correspondence. John Nash used the Kakutani fixed-point theorem to develop this idea, which later earned him a Nobel prize. In physics, fixed points are essential to the theory of phase transitions. Kenneth Wilson used the concept of linearization near unstable fixed points to invent the renormalization group, also earning a Nobel prize. These concepts help explain critical phenomena in the physical world.

Modern technology and computer science rely heavily on these mathematical foundations. Programming language compilers use fixed-point computations for data-flow analysis to optimize code. In type theory, fixed-point combinators allow for the definition of recursive functions in untyped lambda calculus. Even the way we navigate the internet is connected to this idea. The PageRank values used by search engines are the fixed point of a linear transformation based on web links. Similarly, the stationary distribution of a Markov chain is the fixed point of its transition probability function.