Editing Fixed point (mathematics) (section)

==Applications==
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In many fields, equilibria or [[stability theory|stability]] are fundamental concepts that can be described in terms of fixed points.  Some examples follow.

*  In [[projective geometry]], a fixed point of a [[projectivity]] has been called a '''double point'''.<ref>{{cite book |author-link=H. S. M. Coxeter |first=H. S. M. |last=Coxeter |year=1942 |title=Non-Euclidean Geometry |page=36 |publisher=[[University of Toronto Press]] }}</ref><ref>[[G. B. Halsted]] (1906) ''Synthetic Projective Geometry'', page 27</ref>
* In [[economics]], a [[Nash equilibrium]] of a [[game theory|game]] is a fixed point of the game's [[best response|best response correspondence]]. [[John Forbes Nash Jr.|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 [[phase transition|theory of phase transitions]], ''linearization'' near an ''unstable'' fixed point has led to  [[Kenneth G. Wilson|Wilson]]'s Nobel prize-winning work inventing the [[renormalization group]], and to the mathematical explanation of the term "[[critical phenomenon]]."<ref>{{Cite journal|doi = 10.1103/PhysRevB.4.3174|title = Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture|year = 1971|last1 = Wilson|first1 = Kenneth G.|journal = Physical Review B|volume = 4|issue = 9|pages = 3174–3183|bibcode = 1971PhRvB...4.3174W|doi-access = free}}</ref><ref>{{Cite journal|doi = 10.1103/PhysRevB.4.3184|title = Renormalization Group and Critical Phenomena. II. Phase-Space Cell Analysis of Critical Behavior|year = 1971|last1 = Wilson|first1 = Kenneth G.|journal = Physical Review B|volume = 4|issue = 9|pages = 3184–3205|bibcode = 1971PhRvB...4.3184W|doi-access = free}}</ref>
* [[Programming language]] [[compilers]] use fixed point computations for program analysis, for example in [[data-flow analysis]], which is often required for code [[Optimization (computer science)|optimization]]. They are also the core concept used by the generic program analysis method [[abstract interpretation]].<ref>{{Cite web|url=https://www.di.ens.fr/~cousot/COUSOTpapers/POPL77.shtml|title = P. Cousot & R. Cousot, Abstract interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints}}</ref>
* 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.
* Fixed points are used to finding formulas for [[Iterated function|iterated functions]].