In those branches of mathematics called dynamical systems and ergodic theory, the concept of a wandering set formalizes a certain idea of movement and 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 Birkhoff in 1927.
Other articles related to "wandering set, wandering, set":
... case, the whole manifold M is hyperbolic (although it is an open question whether the non-wandering set Ω(f) constitutes the whole M) ... Rufus Bowen showed that the non-wandering set Ω(f) of any axiom A diffeomorphism supports a Markov partition ... The density of the periodic points in the non-wandering set implies its local maximality there exists an open neighborhood U of Ω(f) such that ...
... A wandering set is a collection of wandering points ... More precisely, a subset W of is a wandering set under the action of a discrete group if W is measurable and if, for any the intersection 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 ...
Famous quotes containing the words set and/or wandering:
“Colleges, in like manner, have their indispensable office,to teach elements. But they can only highly serve us, when they aim not to drill, but to create; when they gather from far every ray of various genius to their hospitable halls, and, by the concentrated fires, set the hearts of their youth on flame.”
—Ralph Waldo Emerson (18031882)
“It is a thorny undertaking, and more so than it seems, to follow a movement so wandering as that of our mind, to penetrate the opaque depths of its innermost folds, to pick out and immobilize the innumerable flutterings that agitate it. And it is a new and extraordinary amusement, which withdraws us from the ordinary occupations of the world, yes, even from those most recommended.”
—Michel de Montaigne (15331592)