Applications
In many fields, equilibria or stability are fundamental concepts that can be described in terms of fixed points. For example, in economics, a Nash equilibrium of a game is a fixed point of the game's best response correspondence. However, 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".
In compilers, fixed point computations are used for whole program analysis, which are often required to do code optimization.
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.
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 which remains undefined for problematic sentences like "This sentence is not true"), by recursively defining "truth" starting from the segment of a language which contains no occurrences of the word, and continuing until the process ceases to yield any newly well-defined sentences. (This will take a denumerable infinity of steps.) That is, for a language L, let 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-prime 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 which contains its own truth predicate.
The concept of fixed point can be used to define the convergence of a function.
Read more about this topic: Fixed Point (mathematics)