en>Yobot: WP:CHECKWIKI error fixes - Replaced endash with hyphen in sortkey per WP:MCSTJR using AWB (9100)

2013-04-20T20:39:06Z

WP:CHECKWIKI error fixes - Replaced endash with hyphen in sortkey per WP:MCSTJR using AWB (9100)

← Older revision		Revision as of 22:39, 20 April 2013
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	I ~~like Leaf collecting~~ and ~~pressing~~. ~~Seems boring? Not!~~<br>~~I try~~ to ~~learn Japanese in my spare time.~~<br><br>~~Also visit my web site:~~ [http://~~brokerltd~~.~~com~~/~~adv_vertical_network~~.~~html vertical marketing network~~]		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 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 '''Z'''<nowiki>[[</nowiki>''t''<nowiki>]]</nowiki> 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 ''D''<sub>''k''</sub> is either +1 or −1, according to whether the (''k'' + 1)st iterate ''f''<sup> ''k''+1</sup> has [[local maximum]] or local minimum at ''c''.

			== See also ==

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

			== References ==

			* [[John Milnor]] and [[William Thurston]], ''On iterated maps of the interval''. Dynamical systems (College Park, MD, 1986–87), Lecture Notes in Math., 1342, 465–563, Springer, Berlin, 1988 {{MathSciNet\|id=0970571}}
			* Chris Preston, ''What you need to know to knead''. Adv. Math. 78 (1989), no. 2, 192–252 {{MathSciNet\|id=1029100}}[http://www.math.uni-bielefeld.de/~preston/rest/dynam/text.php online version 2003]

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

en>Compsonheir: typo

2011-06-15T17:45:00Z

typo

New page

I like Leaf collecting and pressing. Seems boring? Not!<br>I try to learn Japanese in my spare time.<br><br>Also visit my web site: [http://brokerltd.com/adv_vertical_network.html vertical marketing network]

Neumann–Neumann methods - Revision history

en>Yobot: WP:CHECKWIKI error fixes - Replaced endash with hyphen in sortkey per WP:MCSTJR using AWB (9100)

en>Compsonheir: typo