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121:"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
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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
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holds cannot have, by definition, a wandering set of positive measure; and is thus an example of a conservative system.
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applies. Intuitively, the connection between wandering sets and dissipation is easily understood: if a portion of the
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Similar definitions follow for the continuous-time and discrete and continuous group actions.
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105:. When a dynamical system has a wandering set of non-zero measure, then the system is a
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can be decomposed into an invariant conservative set and an invariant wandering set.
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In mathematics, a concept that formalizes a certain idea of movement and mixing
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A wandering set is a collection of wandering points. More precisely, a subset
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A common, discrete-time definition of wandering sets starts with a map
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1489:{\displaystyle W^{*}=\bigcup _{\gamma \in \Gamma }\;\;\gamma W.}
1696:, De Gruyter Studies in Mathematics, vol. 6, de Gruyter,
1415:. If there is no such wandering set, the action is said to be
625:{\displaystyle \varphi _{t+s}=\varphi _{t}\circ \varphi _{s}.}
304:
A handier definition requires only that the intersection have
15:
1678:
Alexandre I. Danilenko and Cesar E. Silva (8 April 2009).
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These simpler definitions may be fully generalized to the
756:{\displaystyle \mu \left(\varphi _{t}(U)\cap U\right)=0.}
1050:{\displaystyle \mu \left(\gamma \cdot U\cap U\right)=0}
493:. Similarly, a continuous-time system will have a map
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916:{\displaystyle \{\gamma \cdot x:\gamma \in \Gamma \}}
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457:{\displaystyle \mu \left(f^{n}(U)\cap U\right)=0,}
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536:of the system, with the time-evolution operator
43:but its sources remain unclear because it lacks
308:. To be precise, the definition requires that
850:be a group acting on that set. Given a point
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294:{\displaystyle f^{n}(U)\cap U=\varnothing .}
1681:Ergodic theory: Nonsingular transformations
699:, the time-evolved map is of measure zero:
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101:formalizes a certain idea of movement and
1664:. Cambridge: Cambridge University Press.
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1529:of positive measure, such that the orbit
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1432:Define the trajectory of a wandering set
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74:Learn how and when to remove this message
1425:. For example, any system for which the
1308:{\displaystyle \gamma \in \Gamma -\{e\}}
1130:is non-wandering if, for every open set
811:{\displaystyle \Omega =(X,\Sigma ,\mu )}
1104:is the opposite. In the discrete case,
285:
1221:Wandering sets and dissipative systems
1660:The Ergodic Theory of Discrete Groups
1253:under the action of a discrete group
7:
1085:{\displaystyle \gamma \in \Gamma -V}
525:{\displaystyle \varphi _{t}:X\to X}
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635:In such a case, a wandering point
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1404:{\displaystyle (\Omega ,\Gamma )}
556:being a one-parameter continuous
1525:if there exists a wandering set
347:{\displaystyle (X,\Sigma ,\mu )}
20:
971:if there exists a neighborhood
818:be a measure space, that is, a
532:defining the time evolution or
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1340:{\displaystyle \gamma W\cap W}
1277:is measurable and if, for any
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1609:{\displaystyle \Omega -W^{*}}
960:{\displaystyle x\in \Omega }
869:{\displaystyle x\in \Omega }
109:. This is the opposite of a
1656:Nicholls, Peter J. (1989).
1644:No wandering domain theorem
1632:non-singular transformation
1427:Poincaré recurrence theorem
1379:, and the dynamical system
115:Poincaré recurrence theorem
1746:
1619:is a set of measure zero.
1350:is a set of measure zero.
661:will have a neighbourhood
1692:Krengel, Ulrich (1985),
673:such that for all times
549:{\displaystyle \varphi }
316:, i.e. part of a triple
158:{\displaystyle f:X\to X}
29:This article includes a
1573:{\displaystyle \Omega }
1512:{\displaystyle \Gamma }
1366:{\displaystyle \Gamma }
1266:{\displaystyle \Gamma }
1242:{\displaystyle \Omega }
996:{\displaystyle \Gamma }
843:{\displaystyle \Gamma }
370:{\displaystyle \Sigma }
213:and a positive integer
58:more precise citations.
1610:
1574:
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1522:completely dissipative
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1421:, and the system is a
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1142:> 0, there is some
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1123:{\displaystyle x\in X}
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692:{\displaystyle t>T}
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654:{\displaystyle x\in X}
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486:{\displaystyle n>N}
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236:{\displaystyle n>N}
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191:{\displaystyle x\in X}
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1686:Arxiv arXiv:0803.2424
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1549:{\displaystyle W^{*}}
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1096:Non-wandering points
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390:{\displaystyle \mu }
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1423:conservative system
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983:of the identity in
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111:conservative system
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1624:Hopf decomposition
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217:such that for all
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107:dissipative system
31:list of references
1730:Dynamical systems
1558:almost-everywhere
1459:
1315:the intersection
772:topological group
167:topological space
91:dynamical systems
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828:Borel subsets
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768:group action
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306:measure zero
303:
245:iterated map
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50:Please help
42:
1376:dissipative
1134:containing
1003:such that
941:An element
669:and a time
119:phase space
56:introducing
1725:Limit sets
1714:Categories
1650:References
1150:such that
1138:and every
928:trajectory
876:, the set
397:such that
356:Borel sets
172:. A point
1602:∗
1594:−
1591:Ω
1568:Ω
1560:equal to
1542:∗
1507:Γ
1478:γ
1471:Γ
1468:∈
1465:γ
1461:⋃
1452:∗
1396:Γ
1390:Ω
1361:Γ
1332:∩
1326:γ
1294:−
1291:Γ
1288:∈
1285:γ
1261:Γ
1237:Ω
1188:∩
1161:μ
1115:∈
1077:−
1074:Γ
1071:∈
1068:γ
1031:∩
1025:⋅
1022:γ
1014:μ
991:Γ
955:Ω
952:∈
908:Γ
905:∈
902:γ
893:⋅
890:γ
864:Ω
861:∈
838:Γ
803:μ
797:Σ
782:Ω
737:∩
719:φ
710:μ
646:∈
611:φ
607:∘
598:φ
579:φ
544:φ
517:→
502:φ
435:∩
408:μ
385:μ
365:Σ
339:μ
333:Σ
286:∅
277:∩
183:∈
150:→
125:in 1927.
64:June 2023
1638:See also
1060:for all
467:for all
123:Birkhoff
1630:with a
830:. Let
824:measure
822:with a
52:improve
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103:mixing
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534:flow
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