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en>GeoffreyT2000: /* Examples */

2014-12-25T01:43:00Z

Examples

← Older revision		Revision as of 03:43, 25 December 2014
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	~~In [[mathematics]]~~, ~~the '''no-wandering-domain theorem''~~' is ~~a result on [[dynamical system]]s, proven by [[Dennis Sullivan]] in 1985~~.		I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. To perform domino is something I really enjoy doing. She functions as a journey agent but quickly she'll be on her own. My wife and I live in Mississippi but now I'm considering other choices.<br><br>my website - [http://www.zavodpm.ru/blogs/glennmusserrvji/14565-great-hobby-advice-assist-allow-you-get-going real psychics]

	~~The theorem states that~~ a ~~[[rational function\|rational map]]~~ '~~'f'' : '''Ĉ''' → '''Ĉ''' with deg(''f'') ≥ 2 does not have a [[wandering domain]], where '''Ĉ''' denotes the [[Riemann sphere]]~~. ~~More precisely, for every [[Connected space\|component]] ''U''~~ in ~~the [[Fatou set]] of~~ '~~'f'', the sequence~~

	~~:<math>U,f(U),f(f(U)),\dots,f^n(U), \dots</math>~~

	~~will eventually become periodic~~. ~~Here, ''f''~~<~~sup~~>~~ ''n''~~<~~/sup~~> ~~denotes the~~ [~~[function iteration\|''n''-fold iteration]] of ''f'', that is,~~

	:~~<math>f^n = \underbrace{f \circ f\circ \cdots \circ f}_n~~ .~~</math>~~

	~~[[File:Wandering_domains_for_the_entire_function_f%28z%29%3Dz%2B2πsin%28z%29~~.~~png\|thumb\|alt=An image of the dynamical plane for <math>f(z)=z+2\pi\sin(z)<~~/~~math>.\|This image illustrates the dynamics of <math>f(z)=z+2\pi\sin(z)<~~/~~math>; the Fatou set (consisting entirely of wandering domains) is shown in white, while the Julia set is shown in tones of gray.]]~~


	~~The theorem does not hold for arbitrary maps; for example, the [[Transcendental function\|transcendental map]] <math>f(z)=z+2\pi\sin(z)<~~/~~math> has wandering domains. However, the result can be generalized to many situations where the functions naturally belong to a finite~~-~~dimensional parameter space, most notably to transcendental entire and meromorphic functions with a finite number of singular values.~~


	~~==References==~~
	* [[Lennart Carleson]] and Theodore W. Gamelin, ''Complex Dynamics'', Universitext: Tracts in Mathematics, [[Springer-~~Verlag]], New York, 1993, ISBN 0~~-~~387~~-~~97942~~-~~5 {{MathSciNet\| id=1230383}}~~
	* Dennis Sullivan, ''Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-~~Julia problem on wandering domains'', [[Annals of Mathematics]] 122 (1985), no. 3, 401–18. {{MathSciNet\| id=0819553}}~~
	* S. Zakeri, ''[http://www.math.qc.edu/~zakeri/notes/wander.pdf Sullivan's proof of Fatou's no wandering domain conjecture]''

	~~{{mathapplied~~-~~stub}}~~

	~~[[Category:Ergodic theory]]~~
	~~[[Category:Limit sets]]~~
	~~[[Category:Theorems in dynamical systems]]~~
	~~[[Category:Complex dynamics]~~]

Deltahedron: /* Exact (effective) categories */ cite Pedicchio & Tholen (2004)

2013-03-29T17:50:23Z

Exact (effective) categories: cite Pedicchio & Tholen (2004)

← Older revision		Revision as of 19:50, 29 March 2013
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	~~Wilber Berryhill~~ is the ~~title his mothers and fathers gave him and he totally digs that title~~. ~~To play lacross is one~~ of the ~~issues she loves most~~. ~~Distributing manufacturing~~ is ~~how he makes a living~~. ~~Ohio~~ is ~~exactly where my house~~ is ~~but my husband desires us to move~~.<br><br>~~Have~~ a ~~look at my homepage~~ ... [http://~~myfusionprofits~~.~~com~~/~~groups~~/~~find~~-~~out-about-self-improvement-tips-here/ live psychic reading~~]		In [[mathematics]], the '''no-wandering-domain theorem''' is a result on [[dynamical system]]s, proven by [[Dennis Sullivan]] in 1985.

			The theorem states that a [[rational function\|rational map]] ''f'' : '''Ĉ''' → '''Ĉ''' with deg(''f'') ≥ 2 does not have a [[wandering domain]], where '''Ĉ''' denotes the [[Riemann sphere]]. More precisely, for every [[Connected space\|component]] ''U'' in the [[Fatou set]] of ''f'', the sequence

			:<math>U,f(U),f(f(U)),\dots,f^n(U), \dots</math>

			will eventually become periodic. Here, ''f''<sup> ''n''</sup> denotes the [[function iteration\|''n''-fold iteration]] of ''f'', that is,

			:<math>f^n = \underbrace{f \circ f\circ \cdots \circ f}_n .</math>

			[[File:Wandering_domains_for_the_entire_function_f%28z%29%3Dz%2B2πsin%28z%29.png\|thumb\|alt=An image of the dynamical plane for <math>f(z)=z+2\pi\sin(z)</math>.\|This image illustrates the dynamics of <math>f(z)=z+2\pi\sin(z)</math>; the Fatou set (consisting entirely of wandering domains) is shown in white, while the Julia set is shown in tones of gray.]]


			The theorem does not hold for arbitrary maps; for example, the [[Transcendental function\|transcendental map]] <math>f(z)=z+2\pi\sin(z)</math> has wandering domains. However, the result can be generalized to many situations where the functions naturally belong to a finite-dimensional parameter space, most notably to transcendental entire and meromorphic functions with a finite number of singular values.


			==References==
			* [[Lennart Carleson]] and Theodore W. Gamelin, ''Complex Dynamics'', Universitext: Tracts in Mathematics, [[Springer-Verlag]], New York, 1993, ISBN 0-387-97942-5 {{MathSciNet\| id=1230383}}
			* Dennis Sullivan, ''Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains'', [[Annals of Mathematics]] 122 (1985), no. 3, 401–18. {{MathSciNet\| id=0819553}}
			* S. Zakeri, ''[http://www.math.qc.edu/~zakeri/notes/wander.pdf Sullivan's proof of Fatou's no wandering domain conjecture]''

			{{mathapplied-stub}}

			[[Category:Ergodic theory]]
			[[Category:Limit sets]]
			[[Category:Theorems in dynamical systems]]
			[[Category:Complex dynamics]]

en>LokiClock: "Kernel pair" is here

2012-04-27T15:02:04Z

"Kernel pair" is here

New page

Wilber Berryhill is the title his mothers and fathers gave him and he totally digs that title. To play lacross is one of the issues she loves most. Distributing manufacturing is how he makes a living. Ohio is exactly where my house is but my husband desires us to move.<br><br>Have a look at my homepage ... [http://myfusionprofits.com/groups/find-out-about-self-improvement-tips-here/ live psychic reading]