View source for Milnor–Thurston kneading theory

{{Short description|Mathematical theory in topological dynamics}}
The '''Milnor–Thurston kneading theory''' is a [[mathematics|mathematical]] theory which analyzes the iterates of piecewise monotone [[map (mathematics)|mappings]] of an [[interval (mathematics)|interval]] into itself. The emphasis is on understanding the properties of the mapping that are invariant under [[topological conjugacy]].

The theory had been developed by [[John Milnor]] and [[William Thurston]] in two widely circulated and influential [[Princeton University|Princeton]] preprints from 1977 that were revised in 1981 and finally published in 1988. Applications of the theory include piecewise linear models, counting of [[fixed point (mathematics)|fixed points]], computing the [[total variation]], and constructing an [[invariant measure]] with maximal [[entropy]].

== Short description ==

Kneading theory provides an effective calculus for describing the qualitative behavior of the [[iterated function|iterates]] of a piecewise [[monotonic function|monotone]] mapping ''f'' of a [[closed interval]] ''I'' of the [[real line]] into itself. Some quantitative invariants of this [[discrete dynamical system]], such as the ''lap numbers'' of the iterates and the [[Artin–Mazur zeta function]] of ''f'' are expressed in terms of certain [[matrix (mathematics)|matrices]] and [[formal power series]]. 

The basic invariant of ''f'' is its '''kneading matrix''', a rectangular matrix with coefficients in the ring <math>\Z[[t]]</math> of integer formal power series. A closely related '''kneading determinant''' is a formal power series 

: <math> D(t) = 1 + D_1 t + D_2 t^2 + \cdots\, </math>

with odd integer coefficients. In the simplest case when the map is [[unimodal function|unimodal]], with a maximum at ''c'', each coefficient <math>D_k</math> is either <math>+1</math> or <math>-1</math>, according to whether the <math>(k+1)</math>th iterate <math>f^{k+1}</math> has [[local maximum]]  or local minimum at ''c''.

== See also ==

* [[Sharkovsky theorem]]
* [[Topological entropy]]

== References ==

* {{citation|first1=John W.| last1=Milnor| authorlink1=John Milnor| first2=William| last2=Thurston| authorlink2=William Thurston|contribution=On iterated maps of the interval| title=Dynamical systems (College Park, MD, 1986–87)|series= Lecture Notes in Mathematics|volume= 1342|pages= 465–563| publisher=Springer|location= Berlin|year= 1988|mr=0970571|doi=10.1007/BFb0082847|doi-access=}}
* {{citation|first=Chris|last=Preston|title=What you need to know to knead|journal= [[Advances in Mathematics]]|volume=78 |year=1989|issue=2|pages= 192–252|doi=10.1016/0001-8708(89)90033-9| mr=1029100|doi-access=free}}

{{DEFAULTSORT:Milnor-Thurston kneading theory}}
[[Category:Topological dynamics]]