Fixed point (mathematics)

The fixed point iteration x_n+1 = cos x_n with initial value x₁ = −1.

An attractive fixed point of a function f is a fixed point x₀ of f such that for any value of x in the domain that is close enough to x₀, the iterated function sequence

${\displaystyle x,\ f(x),\ f(f(x)),\ f(f(f(x))),\dots }$

converges to x₀. An expression of prerequisites and proof of the existence of such solution is given by the Banach fixed-point theorem.

The natural cosine function ("natural" means in radians, not degrees or other units) has exactly one fixed point, which is attractive. In this case, "close enough" is not a stringent criterion at all—to demonstrate this, start with any real number and repeatedly press the cos key on a calculator (checking first that the calculator is in "radians" mode). It eventually converges to about 0.739085133, which is a fixed point. That is where the graph of the cosine function intersects the line ${\displaystyle y=x}$.[3]

Not all fixed points are attractive. For example, x = 0 is a fixed point of the function f(x) = 2x, but iteration of this function for any value other than zero rapidly diverges. However, if the function f is continuously differentiable in an open neighbourhood of a fixed point x₀, and ${\displaystyle |f\,'(x_{0})|<1}$, attraction is guaranteed.

Attractive fixed points are a special case of a wider mathematical concept of attractors.

An attractive fixed point is said to be a stable fixed point if it is also Lyapunov stable.

A fixed point is said to be a neutrally stable fixed point if it is Lyapunov stable but not attracting. The center of a linear homogeneous differential equation of the second order is an example of a neutrally stable fixed point.

Multiple attractive points can be collected in an attractive fixed set.

In many fields, equilibria or stability are fundamental concepts that can be described in terms of fixed points. Some examples follow.

• In economics, a Nash equilibrium of a game is a fixed point of the game's best response correspondence. John Nash exploited the Kakutani fixed-point theorem for his seminal paper that won him the Nobel prize in economics.

• In physics, more precisely in the theory of phase transitions, linearisation near an unstable fixed point has led to Wilson's Nobel prize-winning work inventing the renormalization group, and to the mathematical explanation of the term "critical phenomenon."[4][5]

• Programming language compilers use fixed point computations for program analysis, for example in data-flow analysis, which is often required for code optimization. They are also the core concept used by the generic program analysis method abstract interpretation.[6]

• In type theory, the fixed-point combinator allows definition of recursive functions in the untyped lambda calculus.

• The vector of PageRank values of all web pages is the fixed point of a linear transformation derived from the World Wide Web's link structure.

• The stationary distribution of a Markov chain is the fixed point of the one step transition probability function.

• Logician Saul Kripke makes use of fixed points in his influential theory of truth. He shows how one can generate a partially defined truth predicate (one that remains undefined for problematic sentences like "This sentence is not true"), by recursively defining "truth" starting from the segment of a language that contains no occurrences of the word, and continuing until the process ceases to yield any newly well-defined sentences. (This takes a countable infinity of steps.) That is, for a language L, let L′ (read "L-prime") be the language generated by adding to L, for each sentence S in L, the sentence "S is true." A fixed point is reached when L′ is L; at this point sentences like "This sentence is not true" remain undefined, so, according to Kripke, the theory is suitable for a natural language that contains its own truth predicate.

A topological space ${\displaystyle X}$ is said to have the fixed point property (briefly FPP) if for any continuous function

${\displaystyle f\colon X\to X}$

there exists ${\displaystyle x\in X}$ such that ${\displaystyle f(x)=x}$.

The FPP is a topological invariant, i.e. is preserved by any homeomorphism. The FPP is also preserved by any retraction.

According to the Brouwer fixed-point theorem, every compact and convex subset of a Euclidean space has the FPP. Compactness alone does not imply the FPP and convexity is not even a topological property so it makes sense to ask how to topologically characterize the FPP. In 1932 Borsuk asked whether compactness together with contractibility could be a necessary and sufficient condition for the FPP to hold. The problem was open for 20 years until the conjecture was disproved by Kinoshita who found an example of a compact contractible space without the FPP.[7]

The notion and terminology is generalized to a partial order. Let ≤ be a partial order over a set X and let f: X → X be a function over X. Then a prefixpoint (also spelled pre-fixpoint) of f is any p such that f(p) ≤ p. Analogously a postfixpoint (or post-fixpoint) of f is any p such that p ≤ f(p).[8] One way to express the Knaster–Tarski theorem is to say that a monotone function on a complete lattice has a least fixpoint that coincides with its least prefixpoint (and similarly its greatest fixpoint coincides with its greatest postfixpoint). Prefixpoints and postfixpoints have applications in theoretical computer science.[9]

• An Elegant Solution for Drawing a Fixed Point