Wandering set

Summary

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.[citation needed]

Wandering points

A common, discrete-time definition of wandering sets starts with a map of a topological space X. A point is said to be a wandering point if there is a neighbourhood U of x and a positive integer N such that for all , the iterated map is non-intersecting:

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 of Borel sets and a measure such that

for all . Similarly, a continuous-time system will have a map defining the time evolution or flow of the system, with the time-evolution operator being a one-parameter continuous abelian group action on X:

In such a case, a wandering point will have a neighbourhood U of x and a time T such that for all times , the time-evolved map is of measure zero:

These simpler definitions may be fully generalized to the group action of a topological group. Let be a measure space, that is, a set with a measure defined on its Borel subsets. Let be a group acting on that set. Given a point , the set

is called the trajectory or orbit of the point x.

An element is called a wandering point if there exists a neighborhood U of x and a neighborhood V of the identity in such that

for all .

Non-wandering points

A non-wandering point is the opposite. In the discrete case, is non-wandering if, for every open set U containing x and every N > 0, there is some n > N such that

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 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. If there exists a wandering set of positive measure, then the action of is said to be dissipative, and the dynamical system 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

The action of is said to be completely dissipative if there exists a wandering set W of positive measure, such that the orbit is almost-everywhere equal to , that is, if

is a set of measure zero.

The Hopf decomposition states that every measure space with a non-singular transformation can be decomposed into an invariant conservative set and an invariant wandering set.

See also

References

  • Nicholls, Peter J. (1989). The Ergodic Theory of Discrete Groups. Cambridge: Cambridge University Press. ISBN 0-521-37674-2.
  • Alexandre I. Danilenko and Cesar E. Silva (8 April 2009). Ergodic theory: Nonsingular transformations; See Arxiv arXiv:0803.2424.
  • Krengel, Ulrich (1985), Ergodic theorems, De Gruyter Studies in Mathematics, vol. 6, de Gruyter, ISBN 3-11-008478-3