en>Archelon: /* The method */ removed nonsensical "in"; additional $tags$

2014-10-20T18:45:33Z

The method: removed nonsensical "in"; additional <math> tags

en>Frinthruit: /* References */

2013-08-24T19:20:30Z

References

← Older revision		Revision as of 21:20, 24 August 2013
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	Hi there, ~~I am Alyson Boon even though it~~ is ~~not~~ the ~~name~~ on ~~my birth certificate~~. He is ~~an info officer~~. ~~Ohio~~ is ~~exactly where my house~~ is ~~but my husband wants us to move~~. ~~To climb~~ is some ~~thing I really enjoy performing~~.<br><br>~~Feel free~~ to ~~visit my homepage; authentic psychic readings (~~[~~https:~~//~~www~~.~~machlitim~~.~~org~~.il/~~subdomain~~/~~megila~~/~~end~~/~~node~~/~~12300 machlitim~~.~~org~~.il])		In those branches of [[mathematics]] called [[dynamical system]]s and [[ergodic theory]], the concept of a '''wandering set''' formalizes a certain idea of movement and [[mixing (mathematics)\|mixing]] in such systems. When a dynamical system has a wandering set of non-zero measure, then the system is a [[dissipative system]]. This is very much the opposite of a [[conservative system]], for which the ideas of the [[Poincaré recurrence theorem]] apply. Intuitively, the connection between wandering sets and dissipation is easily understood: if a portion of the [[phase space]] "wanders away" during normal time-evolution of the system, and is never visited again, then the system is dissipative. The language of wandering sets can be used to give a precise, mathematical definition to the concept of a dissipative system. The notion of wandering sets in phase space was introduced by [[George David Birkhoff\|Birkhoff]] in 1927{{citation needed\|date=November 2010}}.

			==Wandering points==
			A common, discrete-time definition of wandering sets starts with a map <math>f:X\to X</math> of a [[topological space]] ''X''. A point <math>x\in X</math> is said to be a '''wandering point''' if there is a [[neighbourhood (mathematics)\|neighbourhood]] ''U'' of ''x'' and a positive integer ''N'' such that for all <math>n>N</math>, the [[iterated map]] is non-intersecting:

			:<math>f^n(U) \cap U = \varnothing.\,</math>

			A handier definition requires only that the intersection have measure zero. To be precise, the definition requires that ''X'' be a [[measure space]], i.e. part of a triple <math>(X,\Sigma,\mu)</math> of [[Borel set]]s <math>\Sigma</math> and a measure <math>\mu</math> such that

			:<math>\mu\left(f^n(U) \cap U \right) = 0.\,</math>

			Similarly, a continuous-time system will have a map <math>\varphi_t:X\to X</math> defining the time evolution or [[flow (mathematics)\|flow]] of the system, with the time-evolution operator <math>\varphi</math> being a one-parameter continuous [[abelian group]] [[group action\|action]] on ''X'':

			:<math>\varphi_{t+s} = \varphi_t \circ \varphi_s.\,</math>

			In such a case, a wandering point <math>x\in X</math> will have a [[neighbourhood (mathematics)\|neighbourhood]] ''U'' of ''x'' and a time ''T'' such that for all times <math>t>T</math>, the time-evolved map is of measure zero:

			:<math>\mu\left(\varphi_t(U) \cap U \right) = 0.\,</math>

			These simpler definitions may be fully generalized to a general [[group (mathematics)\|group]] [[group action\|action]]. Let <math>\Omega=(X,\Sigma,\mu)</math> be a [[measure space]], that is, a [[set (mathematics)\|set]] with a [[measure (mathematics)\|measure]] defined on its [[Borel subset]]s. Let <math>\Gamma</math> be a [[Group (mathematics)\|group]] [[group action\|acting]] on that set. Given a point <math>x \in \Omega</math>, the set

			:<math>\{\gamma \cdot x : \gamma \in \Gamma\}</math>

			is called the [[trajectory]] or [[orbit (group theory)\|orbit]] of the point ''x''.

			An element <math>x \in \Omega</math> is called a '''wandering point''' if there exists a [[neighborhood]] ''U'' of ''x'' and a [[neighborhood]] ''V'' of the identity in <math>\Gamma</math> such that
			:<math>\mu\left(\gamma \cdot U \cap U\right)=0</math>

			for all <math>\gamma \in \Gamma-V</math>.

			==Non-wandering points==
			The definition for a '''non-wandering point''' is in a sense the converse. In the discrete case, <math>x\in X</math> is non-wandering if, for every open set ''U'' containing ''x'', one has that

			:<math>\mu\left(f^n(U)\cap U \right) > 0\,</math>

			for some <math>n \ge N</math> and any <math>N \ge 1</math> arbitrarily large. Similar definitions follow for the continuous-time and discrete and continuous group actions.

			==Wandering sets and dissipative systems==
			A wandering set is a collection of wandering points. More precisely, a subset ''W'' of <math>\Omega</math> is a '''wandering set''' under the action of a discrete group <math>\Gamma</math> if ''W'' is measurable and if, for any <math>\gamma \in \Gamma - \{e\}</math> the intersection

			:<math>\gamma W \cap W\,</math>

			is a set of measure zero.

			The concept of a wandering set is in a sense dual to the ideas expressed in the [[Poincaré recurrence theorem]]. If there exists a wandering set of positive measure, then the action of <math>\Gamma</math> is said to be '''dissipative''', and the [[dynamical system]] <math>(\Omega, \Gamma)</math> is said to be a [[dissipative system]]. If there is no such wandering set, the action is said to be '''conservative''', and the system is a [[conservative system]]. For example, any system for which the Poincaré recurrence theorem holds cannot have, by definition, a wandering set of positive measure; and is thus an example of a conservative system.

			Define the trajectory of a wandering set ''W'' as

			:<math>W^* = \cup_{\gamma \in \Gamma} \;\; \gamma W.</math>

			The action of <math>\Gamma</math> is said to be '''completely dissipative''' if there exists a wandering set ''W'' of positive measure, such that the orbit <math>W^*</math> is [[almost-everywhere]] equal to <math>\Omega</math>, that is, if

			:<math>\Omega - W^*\,</math>

			is a set of measure zero.

			==See also==
			* [[No wandering domain theorem]]

			==References==
			*{{cite book \|first=Peter J. \|last=Nicholls \|title=The Ergodic Theory of Discrete Groups \|year=1989 \|publisher=Cambridge University Press \|location=Cambridge \|isbn=0-521-37674-2 }}

			{{DEFAULTSORT:Wandering Set}}
			[[Category:Ergodic theory]]
			[[Category:Limit sets]]
			[[Category:Dynamical systems]]

en>JRSpriggs: /* With a coordinate condition */ switch to the harmonic coordinate condition for more complete cancellation; and apparently it is used more often.

2012-06-24T10:49:19Z

With a coordinate condition: switch to the harmonic coordinate condition for more complete cancellation; and apparently it is used more often.

New page

Hi there, I am Alyson Boon even though it is not the name on my birth certificate. He is an info officer. Ohio is exactly where my house is but my husband wants us to move. To climb is some thing I really enjoy performing.<br><br>Feel free to visit my homepage; authentic psychic readings ([https://www.machlitim.org.il/subdomain/megila/end/node/12300 machlitim.org.il])

Linearized gravity - Revision history

en>Archelon: /* The method */ removed nonsensical "in"; additional tags

en>Frinthruit: /* References */

en>JRSpriggs: /* With a coordinate condition */ switch to the harmonic coordinate condition for more complete cancellation; and apparently it is used more often.

en>Archelon: /* The method */ removed nonsensical "in"; additional $tags$