In mathematics, a fixed point of a function is an element of the function's domain that is mapped to itself by the function. That is to say, c is a fixed point of the function f if f = c. This means f = fn = c, an important terminating consideration when recursively computing f. A set of fixed points is sometimes called a fixed set. For example, if f is defined on the real numbers by then 2 is a fixed point of f, because f = 2. Not all functions have fixed points: for example, if f is a function defined on the real numbers as f = x +1, then it has no fixed points, since x is never equal tox +1 for any real number. In graphical terms, a fixed point x means the point is on the liney = x, or in other words the graph of f has a point in common with that line. Points that come back to the same value after a finite number of iterations of the function are called periodic points. A fixed point is a periodic point with period equal to one. In projective geometry, a fixed point of a projectivity has been called a double point. In Galois theory, the set of the fixed points of a set of field automorphisms is a field called the fixed field of the set of automorphisms.
Attractive fixed points
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 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 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. It eventually converges to about 0.739085133, which is a fixed point. That is where the graph of the cosine function intersects the line. Not all fixed points are attractive. For example, x = 0 is a fixed point of the function f = 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, 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.
Applications
In many fields, equilibria or stability are fundamental concepts that can be described in terms of fixed points. Some examples follow.
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."
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, 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. That is, for a language L, let L′ 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.
Generalization to partial orders: prefixpoint and postfixpoint
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 of f is any p such that f ≤ p. Analogously a postfixpoint of f is any p such that p ≤ f. 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. Prefixpoints and postfixpoints have applications in theoretical computer science.
