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132:"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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116:. 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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1500:{\displaystyle W^{*}=\bigcup _{\gamma \in \Gamma }\;\;\gamma W.}
1707:, De Gruyter Studies in Mathematics, vol. 6, de Gruyter,
1426:. If there is no such wandering set, the action is said to be
636:{\displaystyle \varphi _{t+s}=\varphi _{t}\circ \varphi _{s}.}
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A handier definition requires only that the intersection have
26:
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Alexandre I. Danilenko and Cesar E. Silva (8 April 2009).
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These simpler definitions may be fully generalized to the
767:{\displaystyle \mu \left(\varphi _{t}(U)\cap U\right)=0.}
1061:{\displaystyle \mu \left(\gamma \cdot U\cap U\right)=0}
504:. Similarly, a continuous-time system will have a map
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468:{\displaystyle \mu \left(f^{n}(U)\cap U\right)=0,}
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547:of the system, with the time-evolution operator
54:but its sources remain unclear because it lacks
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861:be a group acting on that set. Given a point
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305:{\displaystyle f^{n}(U)\cap U=\varnothing .}
1692:Ergodic theory: Nonsingular transformations
710:, the time-evolved map is of measure zero:
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112:formalizes a certain idea of movement and
1675:. Cambridge: Cambridge University Press.
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1540:of positive measure, such that the orbit
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1443:Define the trajectory of a wandering set
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85:Learn how and when to remove this message
1436:. For example, any system for which the
1319:{\displaystyle \gamma \in \Gamma -\{e\}}
1141:is non-wandering if, for every open set
822:{\displaystyle \Omega =(X,\Sigma ,\mu )}
1115:is the opposite. In the discrete case,
296:
1232:Wandering sets and dissipative systems
1671:The Ergodic Theory of Discrete Groups
1264:under the action of a discrete group
7:
1096:{\displaystyle \gamma \in \Gamma -V}
536:{\displaystyle \varphi _{t}:X\to X}
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1415:{\displaystyle (\Omega ,\Gamma )}
567:being a one-parameter continuous
1536:if there exists a wandering set
358:{\displaystyle (X,\Sigma ,\mu )}
31:
982:if there exists a neighborhood
829:be a measure space, that is, a
543:defining the time evolution or
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1351:{\displaystyle \gamma W\cap W}
1288:is measurable and if, for any
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1620:{\displaystyle \Omega -W^{*}}
971:{\displaystyle x\in \Omega }
880:{\displaystyle x\in \Omega }
120:. This is the opposite of a
1667:Nicholls, Peter J. (1989).
1655:No wandering domain theorem
1643:non-singular transformation
1438:Poincaré recurrence theorem
1390:, and the dynamical system
126:Poincaré recurrence theorem
1757:
1630:is a set of measure zero.
1361:is a set of measure zero.
672:will have a neighbourhood
1703:Krengel, Ulrich (1985),
684:such that for all times
560:{\displaystyle \varphi }
327:, i.e. part of a triple
169:{\displaystyle f:X\to X}
40:This article includes a
1584:{\displaystyle \Omega }
1523:{\displaystyle \Gamma }
1377:{\displaystyle \Gamma }
1277:{\displaystyle \Gamma }
1253:{\displaystyle \Omega }
1007:{\displaystyle \Gamma }
854:{\displaystyle \Gamma }
381:{\displaystyle \Sigma }
224:and a positive integer
69:more precise citations.
1621:
1585:
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1533:completely dissipative
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1432:, and the system is a
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1153:> 0, there is some
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1134:{\displaystyle x\in X}
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703:{\displaystyle t>T}
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665:{\displaystyle x\in X}
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497:{\displaystyle n>N}
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247:{\displaystyle n>N}
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202:{\displaystyle x\in X}
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1697:Arxiv arXiv:0803.2424
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1560:{\displaystyle W^{*}}
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1107:Non-wandering points
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401:{\displaystyle \mu }
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1434:conservative system
1113:non-wandering point
994:of the identity in
990:and a neighborhood
122:conservative system
108:, the concept of a
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1635:Hopf decomposition
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118:dissipative system
42:list of references
1741:Dynamical systems
1569:almost-everywhere
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1326:the intersection
783:topological group
178:topological space
102:dynamical systems
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16:(Redirected from
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317:measure zero
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256:iterated map
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61:Please help
53:
1387:dissipative
1145:containing
1014:such that
952:An element
680:and a time
130:phase space
67:introducing
1736:Limit sets
1725:Categories
1661:References
1161:such that
1149:and every
939:trajectory
887:, the set
408:such that
367:Borel sets
183:. A point
1613:∗
1605:−
1602:Ω
1579:Ω
1571:equal to
1553:∗
1518:Γ
1489:γ
1482:Γ
1479:∈
1476:γ
1472:⋃
1463:∗
1407:Γ
1401:Ω
1372:Γ
1343:∩
1337:γ
1305:−
1302:Γ
1299:∈
1296:γ
1272:Γ
1248:Ω
1199:∩
1172:μ
1126:∈
1088:−
1085:Γ
1082:∈
1079:γ
1042:∩
1036:⋅
1033:γ
1025:μ
1002:Γ
966:Ω
963:∈
919:Γ
916:∈
913:γ
904:⋅
901:γ
875:Ω
872:∈
849:Γ
814:μ
808:Σ
793:Ω
748:∩
730:φ
721:μ
657:∈
622:φ
618:∘
609:φ
590:φ
555:φ
528:→
513:φ
446:∩
419:μ
396:μ
376:Σ
350:μ
344:Σ
297:∅
288:∩
194:∈
161:→
136:in 1927.
75:June 2023
1649:See also
1071:for all
478:for all
134:Birkhoff
1641:with a
841:. Let
835:measure
833:with a
63:improve
1711:
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572:action
254:, the
114:mixing
1260:is a
1157:>
943:orbit
781:of a
323:be a
176:of a
48:, or
1709:ISBN
1677:ISBN
1633:The
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695:>
545:flow
489:>
239:>
104:and
1567:is
1447:as
1284:if
1240:of
986:of
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831:set
676:of
574:on
365:of
220:of
100:In
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1213:0.
1111:A
1103:.
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578::
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1699:.
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1398:(
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1314:}
1311:e
1308:{
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1206:)
1202:U
1196:)
1193:U
1190:(
1185:n
1181:f
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1159:N
1155:n
1151:N
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1056:0
1053:=
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960:x
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922:}
910::
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898:{
869:x
817:)
811:,
805:,
802:X
799:(
796:=
759:=
755:)
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678:x
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626:s
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600:s
597:+
594:t
576:X
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522::
517:t
492:N
486:n
463:,
460:0
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449:U
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440:U
437:(
432:n
428:f
423:(
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341:,
338:X
335:(
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300:.
294:=
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282:U
279:(
274:n
270:f
242:N
236:n
226:N
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218:U
197:X
191:x
181:X
164:X
158:X
155::
152:f
88:)
82:(
77:)
73:(
59:.
20:)
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