Fixed point (mathematics)

Imagine a rule that moves things. Most things will change. But one spot stays still. This spot is a fixed point.

A fixed point does not change. It stays in the same place. Some rules have many fixed points. Other rules have none at all.

We can use math to find them. We can follow a path to find a spot. This path might lead us to the fixed point.

Some rules are special. They always have at least one fixed point. This can help us find balance. It can help us find where things stay steady.

Imagine a rule that moves things. Most things will change. But one spot stays still. This spot is a fixed point.

A fixed point is a value that stays the same. If you apply a rule to it, the result is the same value. Some rules have many fixed points. Other rules have none at all. For example, a rule might always add one. That rule has no fixed point because the number always changes.

We can use a set of steps to find them. This is called fixed-point iteration. We pick a starting number and follow the rule. We keep doing this over and over. We hope the numbers lead us to a fixed point.

Some math rules are very special. They guarantee that a fixed point exists. The Brouwer fixed-point theorem is one such rule. It says a certain type of rule must have a fixed point. This helps us find balance in many areas. In economics, it helps find a stable state. In physics, it helps explain how things change. Even web searches use fixed points to rank pages.

Imagine a rule that changes numbers or shapes. Most things will move or change when the rule is applied. But a fixed point is a special value that stays exactly the same. In math, we call this an invariant point. If you apply a function to a fixed point, you get that same point back. Some rules have many fixed points, while others have none at all. For example, a rule that always adds one has no fixed point. This is because every number will always change to something else.

We can find these points using a method called fixed-point iteration. This is a way to compute fixed points in numerical analysis. You start with a single number and apply the rule over and over. This creates a sequence of new numbers. We hope this sequence will converge, or move closer, to a fixed point. If the function is continuous, we can often prove this works. Mathematicians also study if a point is attracting or repelling. An attracting point pulls others toward it during the process.

Many famous mathematicians have studied these special points. In 1911, L.E.J. Brouwer shared a major theorem. The Brouwer fixed-point theorem says a continuous function must have a fixed point under certain conditions. It does not tell us how to find it, though. Later, in 1922, Stefan Banach provided a different rule. The Banach fixed-point theorem gives a way to guarantee that iteration will work. Other experts like Lefschetz and Nielsen created ways to count these points. These ideas help us understand the shape of math.

Fixed points appear in many different areas of science. In economics, John Nash used them to find a stable state. This work helped him win a Nobel prize. In physics, fixed points help explain phase transitions. This led to Kenneth Wilson winning a Nobel prize too. Even the internet uses this idea to work. The PageRank values for web pages are a fixed point. This helps search engines rank which pages are most important.

These ideas connect to things you might see in school. In algebra, fixed points can belong to groups or rings. In computer science, they help with program analysis. They even help define recursive functions in logic. Some math rules use fixed points to find balance. This is called an equilibrium in many fields. Whether in a game or a physics lab, fixed points show us where things stay still. They help us find order in a changing world.

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.